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THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 


PRESENTED  BY 

PROF.  CHARLES  A.  KOFOID  AND 

MRS.  PRUDENCE  W.  KOFOID 


THE 


ELEMENTS 


DIFFERENTIAL  CALCULUS; 


COMPREHENDING  THE 


GENERAL  THEORY  OF  CURVE  SURFACES, 


CURVES  OF  DOUBLE  CURVATURE. 


INTKNDID   FOR  THB   USE    Or 


MATHEMATICAL   STUDENTS    IN    SCHOOLS    AND   UNIVERSITIES. 


BY  J.  R.  YOUNG, 

AUTHOR  or 
THE  ELEMENTS  OF  ANALYTICAL  CEOMETRT.' 


REVISED  AND  CORRECTED,  BT 

MICHAEL  O'SHANNESSY,  A.M. 


CAREY,   LEA  &  BLANCHARD, 

CHESNUT-STREET. 

1833. 


"Entered  according  to  Act  of  Congress,  the  6th  of  March,  in  the  year  1833, 
by  G.  F.  Hopkins  &  Son,  in  the  office  of  the  Clerk  of  the  Southern  District  of 
New- York." 


PrinWd  by  U.  F.  UOPKINa  &  s6n,  Now.york. 


j^m 


ADVERTISEMENT. 


This  edition  of  Young's  Differential  and  Integral  Calculus 
is  presented  to  the  American  public,  with  a  confidence  in  its 
favourable  reception,  proportionate  to  that  which  the  original 
acquired  in  England.  The  text  has  not  been  materially  al- 
tered, though  many  errors  have  been  corrected,  some  of  which 
by  Professor  Dodd  of  Princeton  College,  N.  J. 

These  volumes  will  be  found  to  contain  a  full  elementary 
course  of  the  subject  of  which  they  treat,  and  well  adapted  as 
a  text  book  for  Colleges  and  Universities. 

The  second  volume,  treating  exclusively  of  the  Integral 
Calculus,  is  now  in  press,  and  will  be  speedily  published. 

New- York,  March,  1833. 


Digitized  by  tine  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementsofdifferOOyounrich 


PREFACE 


The  object  of  the  present  volume  is  to  teach  the  principles  of  the 
Differential  Calculus,  and  to  show  the  application  of  these  principles 
to  several  interesting  and  important  inquiries,  more  particularly  to  the 
general  theory  of  Curves  and  Surfaces.  Throughout  these  applica- 
tions I  have  endeavoured  to  preserve  the  strictest  rigour  in  the  varioiis 
processes  employed,  so  that  the  student  who  may  have  hitherto  been 
accustomed  only  to  the  pdre  reasoning  of  the  ancient  geometry  will 
not,  I  think,  find  in  these  higher  order  of  researches  any  principle 
adopted,  or  any  assumption  made,  inconsistent  with  his  previous  no- 
tions of  mathematical  accuracy.  If  I  have,  indeed,  succeeded  in 
accomplishing  this  very  desirable  object,  and  have  really  shown 
that  the  applications  of  the  Calculus  do  not  necessarily  involve  any 
principle  that  will  not  bear  the  most  scrupulous  examination,  I  may, 
perhaps,  be  allowed  to  think  that  I  have,  in  this  small  volume,  con- 
tributed a  little  towards  the  perfecting  of  the  most  powerful  instru- 
ment which  the  modern  analysis  places  in  the  hand  of  the  mathema- 
tician. 

It  is  the  adoption  of  exceptionable  principles,  and  even,  in  some 
cases,  of  contradictory  theories,  into  the  elements  of  this  science, 
that  have  no  doubt  been  the  chief  causes  why  it  has  hitherto  been  so 
little  studied  in  a  country  where  the  ancient  geometry  has  been  so 
extensively  and  so  successfully  cultivated.  The  student  who  pro- 
ceeds from  the  works  of  Euclid  or  of  Apollonius  to  study  those  of  our 
modern  analysts,  will  be  naturally  enough  startled  to  find  that  in  the 
theory  of  the  Differential  Calculus  he  is  to  consider  that  as  absolutely 
nothing  which,  in  the  application  of  that  theory,  is  to  be  considered 
a  quantity  infinitely  small.  He  will  naturally  enough  be  startled  to 
find  that  a  conclusion  is  to  be  taken  as  general,  when  he  is  at  the 


VI  PREFACE. 

same  time  told  that  the  process  which  led  to  that  conclusion  has  fail- 
ing cafees ;  and  yet  one  or  both  of  these  inconsistencies  pervade  more 
or  less  every  book  on  the  Calculus  which  I  have  had  an  opportunity 
of  examining. 

The  whole  theory  of  what  the  French  mathematicians  vaguely  call 
consecutive  points  and  consecutive  elements,  involves  the  first  of  these 
objectionable  principles  ;*  for,  if  the  abscissa  of  any  point  be  repre- 
sented by  X,  then  the  abscissa  of  the  consecutive  point,  or  that  sepa- 
rated from  the  former  by  an  infinitely  small  interval,  is  represented 
by  .c  +  d.v,  although  dx,  at  the  outset  of  the  subject,  is  said  to  be  0. 
Again,  the  theory  of  tangents,  the  radius  of  curvature,  principles  of 
osculation,  &c.,  are  all  made  to  depend  upon  Taylor's  theorem,  and 
therefore  can  strictly  apply  only  at  those  points  of  the  curve  where 
this  theorem  does  not  fail :  the  conclusions,  however,  are  to  be  re- 
ceived in  all  their  generality.! 

*  It  is  to  be  rcigrctted  that  terms  so  vague  and  indofinlte  should  be  introduced 
into  the  caraci  sciences;  and  it  is  more  to  be  regretted  that  English  elementary  writers 
should  adopt  them  merely  because  they  arc  used  by  the  French,  and  that  too  with- 
out examining  into  the  import  these  tenns  carry  in  the  works  from  wliichthey  are 
copied.  In  a  recent  production  of  the  University  of  Cambridge,  the  autlior,  in  at- 
tempting to  follow  the  French  mode  of  solving  a  certain  problem,  has  confounded 
consecutive  points  with  consec^itive  elements,  two  very  distinct  things :  although 
neither  very  intellisible,  the  consequence  of  this  mistake  is,  that  the  result  is  not 
what  was  intended;  so  that,  after  the  process  is  fairly  finished,  a  new  counter- 
balancing error  is  introduced,  and  tlius  the  solution  righted ! 

1 1  am  anxious  not  to  be  misunderstood  here,  and  shall  therefore  state  specifi- 
cally the  nature  of  my  objection.  In  establisliing  the  theory  of  contact,  &c.,  bff 
aid  of  Taylor's  theorem,  it  is  assumed  that  a  value  may  be  given  to  the  increment 
h  so  small  as  to  render  the  term  into  which  it  enters  greater  than  all  the  following 
terms  of  the  series  taken  together.  Now  how  can  a  function  of  absolutely  inde- 
terminate quantities  be  shown  to  be  greater  or  less  than  a  series  of  other  functions 
of  the  same  indeterminate  quantities  without,  at  least,  assuming  some  determinate 
relation  among  them?  If  we  say  that  the  assertion  applies,  whatever  particular 
value  we  substitute  for  the  indeterminate  in  the  proposed  functions  or  differential 
coefficients,  we  merely  shift  the  dilemma,  for  an  indefinite  number  of  these  particu- 
lar values  may  render  the  functions  all  infinite;  and  we  shall  be  equally  at  a  loss 
to  conceive  how  one  of  these  infinite  quantities  can  be  greater  or  less  tlian  the 
others.  It  appears,  therefore,  that  the  usual  process  by  which  the  theory  of  con- 
tact is  established,  applies  rigorously  only  to  those  points  of  cur\xs  for  which 
Taylor^s  development  does  not  fail,  and  T  cannot  help  thinking  that  on  these 
grounds  the  .^nalylical  Tliem-y  of  Functions,  by  Lagrange,  in  its  application  to  Ge- 


PREFACE.  Vll 

If  this  statement  be  true,  it  is  not  to  be  wondered  at  that  students 
so  often  abandon  the  study  of  this  science,  less  discouraged  with  its 
difficulties  than  disgusted  with  its  inconsistencies.  To  remove  these 
inconsistencies,  which  so  often  harass  and  impede  the  student's  pro- 
gress, has  been  my  object  in  the  present  volume  ;  and,  aUhough  my 
endeavours  may  not  have  entirely  succeeded,  I  have  still  reason  to 
hope  that  they  have  not  entirely  failed.  The  following  brief  outline 
will  convey  a  notion  of  the  extent  and  pretensions  of  the  book ;  a  more 
detailed  enumeration  of  the  various  topics  treated  of,  will  be  found  in 
the  table  of  contents. 

I  have  taken  for  the  basis  of  the  theory  the  method  of  limits  first 
employed  by  JVeivton,  although  designated  by  foreign  writers  as  the 
method  of  d' Member t.  I  consider  this  method  to  be  as  unexceptiona- 
ble as  that  o(  Lao-range,  and  on  account  of  its  greater  simplicity, 
better  adapted  to  elementary  instruction. 

The  First  Chapter  is  devoted  to  the  exposition  of  the  fundamental 
principles  ;  and  in  explaining  the  notation  I  have  been  careful  to  im- 
press upon  the  student's  mind  that  tlie  differentials  dx,  dy,  &c.  are  in 
themselves  absolutely  of  no  value,  and  that  their  ratios  only  are  sig- 
nificant :  this  is  tlie  foundation  of  the  whole  theory,  and  it  has  been 
adhered  to  throughout  the  volume,  without  any  shifting  of  the  hypo- 
thesis. 

In  the  Second  Chapter  it  is  shown,  that  i£fx  represent  any  function 
of  X,  and  x  be  changed  into  x  +  /t,  the  new  state  y  (x  +  h)  of  the 
function  may  always  be  developed  according  to  the  ascending  inte- 
gral powers  of  the  increment  h  ;  and  this  leads  to  the  important  con- 
clusion that  the  coefficient  of  the  second  term  in  the  development  of 
the  function  f{x-\-  h)  is  the  diflferential  coefficient  derived  from  the 
function /x;  a  fact  which  Lagrange  has  made  the  foundation  of  his 

ometry  is  defective,  altliough  I  feel  anxious  to  express  my  opinion  of  tliat  celebra- 
ted performance  witii  all  becoming  caution  and  humility.  Indeed  Lagrange  him- 
self has  admitted  this  defect,  and  observes,  {Thiorie  des  FoncHons,  p.  181,)  "  GLuoi- 
que  ccs  exceptions  nc  portent  aucune  atteinte  ci  la  thfeoric  g^nferale,  il  est  n6ces- 
saire,  pour  ne  rien  laisscr  k  desirer,  de  voir  comment  die  doit  etre  modifier  dans 
les  cas  particuliers  dont  il  s'agit"  (See  note  C  at  the  end.)  But  he  has  not 
modified  the  expression  deduced  from  this  exceptionable  thebry  for  the  radius  of 
curvatiue,  which  indeed  is  always  applicable  whether  the  differential  coefficients 
become  infinite  or  not,  although,  for  reasons  already  assigned,  (he  process  which 
led  to  it  restricts  its  application  to  particular  points. 


^111  PREFACE. 

theory  of  analytical  functions.  The  chapter  then  goes  on  to  treat  of 
the  differentiation  of  the  various  kinds  of  functions,  algebraic  and 
transcendental,  direct  and  inverse,  and  concludes  with  an  article  on 
successive  differentiation. 

The  Third  Chapter  is  devoted  to  M.aclaurin' s  theorem,  and  its  ap- 
pHcation  is  shown  in  the  development  of  a  great  variety  of  functions. 
Occasion  is  taken,  in  the  course  of  this  chapter,  to  introduce  to  the 
student's  attention  some  valuable  analytical  formulas  and  expressions 
from  Euler,  Demoivre,  Cotes,  and  other  celebrated  analysts,  together 
with  those  curious  properties  of  the  circle  discovered  by  Co fes  and 
Demoivre. 

The  Fourth  Chapter  is  on  Taylor's  theorem,  which  m.akes  known 
the  actual  development  of  the  function /(a;  +  h)  according  to  the 
form  established  in  the  second  chapter.  From  this  theorem  are  de- 
rived commodious  expressions  for  the  total  differential  coefficient 
when  the  function  is  compUcated,  and  whether  its  form  be  explicit  or 
implicit ;  the  whole  being  illustrated  by  a  variety  of  examples. 

The  Fifth  Chapter  contains  the  complete  theory  of  vanishing  frac- 
tions. 

The  Sixth  is  on  the  maxima  and  minima  values  of  functions  of  a 
single  variable,  and  will,  I  think,  be  found  to  contain  several  original 
remarks  and  improved  processes. 

Chapter  the  Seventh  is  on  the  differentiation  and  development  of 
functions  of  two  independent  variables.  The  usual  method  of  obtain- 
ing the  development  of  a  function  of  two  variables  according  to  the 
powers  of  the  increments,  is  to  develop  first  on  the  supposition  that  x 
only  varies  and  that  y  is  constant,  and  afterwards  to  consider  y,  which 
is  assumed  to  enter  into  the  coefficients,  to  be  changed  into  y  -\-  h. 
But  y  may  be  so  combined  with  x  in  the  function  F  {x,  y)  that  it  shall, 
when  considered  as  a  constant,  disappear  from  all  the  differential  co- 
efficients, which  circumstances  should  be  pointed  out  and  be  shown 
not  to  affect  the  truth  of  the  result :  I  have,  however,  avoided  the  ne- 
cessity of  showing  this,  by  proceeding  rather  differently.  The  chap- 
ter concludes  with  Lagrange's  Theorem,  concisely  demonstrated  and 
applied  to  several  examples. 

The  Eighth  Chapter  completes  the  theory  of  maxima  and  minima, 
by  applying  the  principles  delivered  in  chapter  VI.  to  functions  of  two 
independent  variables,  and  it  also  contains  an  important  article  on 


PREFACE.  IX 

changing  the  independent  variable,  a  subject  very  improperly  omitted 
in  all  the  English  books. 

The  Ninth  Chapter  is  devoted  to  a  matter  of  considerable  import- 
ance, viz.  to  the  examination  of  the  cases  in  which  Taylor's  theorem 
fails ;  and  I  have,  I  thuik,  satisfactorily  shown,  that  these  failing  cases 
are  always  indicated  by  the  differential  coefficients  becoming  infinite, 
and  that  the  theorem  does  not  fail  when  these  coefficients  become 
imaginary,  as  Lacroix,  and  others  after  him,  have  asserted.  Besides 
the  correction  of  this  erroneous  doctrine,  which  has  been  sanctioned 
by  names  of  the  highest  reputation,  another  very  remarkable  over- 
sight, though  of  far  less  importance,  is  detected  in  the  Calcul  dcs 
Fonctions  of  Lagrange,  and  is  pointed  out  in  the  present  chapter  :  it 
has  been  unsuspectingly  copied  by  other  writers  ;  and  thus  an  entirely 
wronflf  solution  to  a  very  simple  problem  has  been  printed,  and  re- 
printed, without  any  examination  into  the  principles  employed  in  it ; 
and  which,  I  suppose,  the  high  reputation  of  Lagrange  was  consider- 
ed to  render  unnecessary. 

These  nine  chapters  constitute  the  First  Section  of  the  work,  and 
comprise  the  pure  theory  of  the  subject ;  the  remaining  part  is  devot- 
ed to  the  application  of  this  to  geometry,  and  is  divided  into  two  parts, 
the  fii-st  containing  the  theory  of  plane  curves,  and  the  second  the 
theory  of  curve  surfaces,  and  of  curves  of  double  curvature. 

The  First  Chapter  in  the  Second  Section  explains  the  method  of  tan- 
gents, and  the  general  differential  equation  of  the  tangent  to  any  plane 
curve  is  obtained  by  the  same  means  that  the  equation  is  obtained  in 
analytical  geometry,  and  is  therefore  independent  of  the  failing  cases  of 
Taylor's  theorem.  The  method  of  tangents  naturally  leads  to  the  con- 
siderationof  rectilinear  asymptotes,  which  is,  therefore,  treated  of  in  this 
chapter,  and  several  examples  are  given,  as  well  when  the  curve  is 
referred  to  polar  as  to  rectangular  coordinates,  and  a  few  passing  ob- 
servations made  on  the  circular  asymptotes  to  spiral  curves,  the  chap- 
ter terminating  with  the  differential  expression  for  the  arc  of  any  plane 
curve  determined  without  the  aid  of  Taylor's  theorem. 

The  Second  Chapter  contauis  the  tlieory  of  osculation,  which  is 
shown  to  be  unaffected  by  the  failing  cases  of  Taylor's  theorem,  al- 
though this  is  employed  to  estabhsh  the  theory.  The  expressions  for 
the  radius  of  curvature  are  afterwards  deduced,  and  several  examples 

B 


X  PREFACE. 

of  their  application  given  principally  to  the  curves  of  the  second  order, 
and  an  instance  of  their  utility  shown  in  determining  the  ratio  of  the 
earth's  diameters. 

The  Third  Chapter  is  on  involutes, evolutes,  and  consecutive  curves, 
and  contains  some  interesting  theorems  and  practical  examples.  Of 
what  the  French  call  consecutive  curves,  I  have  endeavoured  to  give 
a  clear  and  satisfactory  explanation,  unmixed  with  any  vague  notions 
about  infinity. 

The  Fourth  Chapter  is  on  the  singular  points  of  curves,  and  con- 
tains easy  rules  for  detecting  them,  from  an  examination  of  the  equa- 
tion of  the  curve.  This  chapter  also  contains  the  general  theory  of 
curvilinear  asymptotes,  and  completes  the  Second  Section,  or  that 
assigned  to  the  consideration  of  plane  curves. 

The  Third  Section  is  devoted  to  the  general  theory  of  curve  sur- 
faces, and  of  curves  of  double  curvature ;  in  the  First  Chapter  of 
which  are  established  the  several  forms  of  the  equations  of  the  tan- 
gent plane  and  normal  line  at  any  point  of  a  curve  surface,  and  of  the 
linear  tangent  and  normal  plane  at  any  point  of  a  curve  of  double  cur- 
vature. 

In  the  Second  Chapter  the  theory  of  conical  and  cylindrical  surfa- 
ces is  discussed,  as  also  that  of  surfaces  of  revolution ;  and  that  re- 
markable case  is  examined,  where  the  revolution  of  a  straight  line 
produces  the  same  surface  as  the  revolution  of  the  hyperbola,  to  which 
this  line  is  an  asymptote.  Throughout  this  chapter  are  interspersed 
many  valuable  and  mteresting  appUcations  of  the  calculus,  chiefly 
from  Monge.  The  Third  Chapter  embraces  the  theory  of  the  curva- 
ture of  surfaces  in  general,  and  will  be  found  to  form  a  collection  of 
very  beautiful  theorems,  the  results,  principally,  of  the  researches  of 
Euler,  Monge,  and  Bupin.  Most  of  these  theorems  have,  however, 
usually  been  established  by  the  aid  of  the  infinitesimal  calculus,  or  by 
the  use  of  some  other  equally  objectionable  principle ;  they  are  here 
fairly  deduced  from  the  principles  of  the  differential  calculus,  without, 
in  any  instance,  departing  from  those  principles,  as  laid  dovvn  in  the 
preliminary  chapter.  Those  who  are  famiUar  with  these  inquiries  will 
find  that  I  have  obtained  some  of  these  theorems  in  a  manner  much 
more  simple  and  concise  than  has  hitherto  been  done.  I  need  only 
mention  here,  as  instances  of  this  simphcity,  the  theorems  of  Euler 
and  of  JVfewsmer,  at  pages  182  and  186. 


PREFACE.  XI 

The  Fourth  Chapter  is  on  twisted  mrfaces,  a  class  of  surfaces  which 
have  never  been  treated  of,  to  any  extent,  by  any  English  author,  al- 
though, as  has  been  recently  shown,  the  English  were  the  first  who 
noticed  the  peculiarities  of  certain  individual  surfaces  belonging  to 
this  extensive  class.*  For  what  is  here  given,  I  am  indebted  to  the 
French  mathematicians,  to  JVLonge  principally,  and  also  to  the  Che- 
valier Le  Roy,  who  has  recently  published  a  very  neat  and  compre- 
hensive little  treatise  on  curves  and  surfaces. 

The  Fifth  Chapter  treats  on  the  developable  surfaces,  or  those 
which,  like  the  cone  and  cylinder,  may,  if  flexible,  be  unrolled  upon 
a  plane,  without  being  twisted  or  torn.  The  Sixth  Chapter  is  on 
curves  of  double  curvature;  and  the  Seventh,  which  concludes  the 
volume,  contains  a  few  miscellaneous  propositions  intimately  connect- 
ed with  the  theory  of  surfaces.  From  the  foregoing  brief  analysis,  it 
will  appear  evident  to  those  familiar  with  the  present  state  of  mathe- 
matical instruction  in  this  country,  that  I  have  introduced,  into  a  little 
duodecimo  volume,  a  more  comprehensive  view  of  the  theory  and 
applications  of  the  differential  calculus  than  has  yet  appeared  in  the 
English  language.  But  I  have  aimed  at  more  than  this  ;  1  have  en- 
deavoured to  simplify  and  improve  much  that  I  have  adopted  from 
foreign  sources  ;  and,  above  all,  to  estabhsh  every  thing  here  taught, 
upon  principles  free  from  inconsistency  and  logical  objections  ;  and 
if  it  be  found,  upon  examination,  that  1  have  entirely  failed  in  this  en- 
deavour, I  shall  certainly  feel  a  proportionate  disappointment. 

I  am  not,  however,  so  sanguine  as  to  look  for  much  public  en- 
couragement of  my  labours,  however  successfully  they  may  have 
been  devoted  :  it  is  not  customary  to  place  much  value,  in  this  coun- 
try, upon  any  mathematical  production,  of  whatever  merit,  that  does 
not  emanate  from  Cambridge.  The  hereditary  reputation  enjoyed 
by  this  University,  and  bequeathed  to  it  by  the  genius  of  Barrow,  of 
jyeivton,  and  of  Cotes,  seems  to  have  endowed  it  with  such  strong 
claims  on  the  public  attention  and  respect,  that  every  thing  it  puts 
forth  is  always  received  as  the  best  of  its  kind.  If  this  be  the  case 
with  Cambridge  books,  of  course  it  is  also  the  case  with  Cambridge 
men,  and  accordingly  we  find  almost  all  our  public  mathematical 
situations  filled  by  members  of  this  University.     It  is  true  that  now 

*  See  Leybourn^s  Repository,  No.  22. 


Xll  PREFACE. 

and  then,  in  the  course  of  half  a  century,  we  find  an  exception  to  this ; 
one  or  two  instances  on  record  have  undoubtedly  occurred,  where  it 
has  been,  by  some  means  or  other,  discovered  that  men  who  had  ne- 
ver seen  Cambridge  knew  a  little  of  mathematics,  as  in  the  case  of 
Thomas  Simpson,  and  of  Dr.  Hutton ;  but  such  instances  are  rare. 
It  is  not  for  me  to  inquire  into  the  justice  of  this  exclusive  system ; 
but,  while  such  a  system  prevails,  there  need  be  little  wonder  at  the 
decline  of  science  in  England :  while  all  inducement  to  cultivate  sci- 
ence is  thus  confined  to  a  particular  set  of  men,  no  wonder  that  its 
votaries  are  few.  It  is  to  be  hoped,  however,  that  in  the  present 
"  liberal  and  enlightened  age,"  such  a  state  of  things  will  not  long 
continue,  and  that  even  the  poor  and  unfriended  student  may  be  cheer- 
ed up,  amidst  all  the  obstacles  that  surround  him,  in  the  laborious  and 
difficult,  but  subUme  and  elevating  career  on  which  he  has  entered, 
by  a  well-founded  assurance  that  his  exertions,  if  successful,  will  not 
be  the  less  appreciated  because  they  were  solitary  and  unassisted. 

May  12,  1831. 

J.  R.  YOUNG. 


CONTENTS 


SECTION    I. 

On  the  Differentiation  of  Functions  in  general. 

•Article  Page 

1.  A  Function  defined        -           -            -            -           -  -            -1 

2.  Effect  produced  on  the  function  by  a  change  in  the  variable  -      2 

3.  Differential  coefficient  determined          -            -            -  -            -      3 

4.  General  form  of  the  development  of/ (a;  +  A)                -  -            -      5 

5.  The  coefficient  of  the  second  term  in  the  general  development  is  the  dif- 

ferential coefficient  derived  from  the  function /a;         -  -  -      8 

6.  To  differentiate  the  product  of  two  or  more  functions  of  the  same  variable      9 

7.  To  differentiate  a  fraction  -  -  -  -  -  -10 

8.  To  differentiate  any  power  of  a  fimction  -  -  -  -    ib. 

9.  To  differentiate  an  expression  consisting  of  several  functions  of  the  same 

variable         -  -  -  -  -  -  -  -12 

10.  Application  of  the  preceding  rules  to  examples  -  -  -    ib. 

11.  Transcendental  functions  -  -  -  -  -  -15 

12.  To  find  the  differential  of  a  logarithm  -  -  -  -    ib. 

13.  To  differentiate  an  exponential  function  -  -  -  -     16 

Examples  on  transcendental  functions           -            -            -  -  ib. 

14.  To  differentiate  circular  functions          -            -            -            -  -  19 

15.  Differentiation  of  inverse  functions         -            -            -            -  -  21 

16.  Forms  of  the  differentials  when  the  radius  is  arbitrary    -            -  -  24 

17.  Successive  differentiation  explained        -            -            -            -  -  25 

18.  Illustrationsof  the  process          -            -            -            -            -  -  26 

19.  Investigation  of  Maclaurin's  Theorem     -           -            -            -  -  28 

20.  Application  of  Maclaurin's  theorem  to  the  development  of  functions  -  29 

21.  Deduction  of -EuZer's  expressions  for  the  sine  and  cosine  of  an  arc,  by 

means  of  imaginary  exponentials       -  -  -  -  -    31 

22.  Demoivre^s  formula,  and  series  for  the  sine  and  cosine  of  a  multiple  arc  -    32 

23.  Decomposition  of  the  expression!/*"  —  2j/  cos.  0+1  into  its  quadratic 

factors  -  -  -  -  -  -  -  -33 

24.  Demotfre's  property  of  the  circle  -  -  -  -  -    34 

25.  Cotes'*  properties  of  the  circle    -  -  -  -  -  -    35 


XIV  CONTENTS. 

^rtklt  Pag* 

26.  JbAn  BamouHi's  development  of  ( -y/  —  1)       -  *  "  -    ib. 

Developments  of  tan.  x  and  tan.  ~^x  -  •  •  -  36 

27.  Evler^s  series  for  approximating  to  the  circumference  of  a  circle  -  38 

28.  Bertran(Ps  more  convergent  series  -  -         *  -  -  -  ib, 

29.  Examples  for  exercise    -  -  -  -  -  -  -39 

30.  Investigation  of  Taylor's  Theorem  -  -  -  -  -  40 

31.  JV/ocZawrin's  theorem  deduced  from  Taylor's        -  -  -  -  42 

32.  Application  of  Taylor's  theorem  to  the  development  of  fvmctiona  -  ib- 

33.  Of  a  function  of  a  function  of  a  single  variable    -  -  -  -  44 

34.  Examples  of  the  application  of  this  form  -  -  -  -  45 

35.  Form  of  the  differential  coefficient  derived  from  the  function  u  =  Y{p,q,) 

where  ji  and  q  are  functions  of  the  same  variable        -  -  -    ib. 

36.  Form  of  the  coefficient  when  the  function  is  v  =  F  ( p,  g,  r,)       -  -    46 

37.  Distinction  between  partial  and  total  differential  coefficients        -  -    47 

38.  Examples  -..-..--48 

39.  Differentiation  and  development  of  implicit  functions       -  -  -    49 

40.  On  vanishing  fractions    -  -  -  -  -.-  -52 

41.  Application  of  the  calculus  to  determine  the  true  value  of  a  vanishing 

fraction  -  -  -  -  -  -  -  -53 

42.  Determination  of  the  value  when  Taylor's  theorem  fails  -  -    56 

43.  Determination  of  the  value  of  a  fraction,  of  which  both  numerator  and  de- 

nominator are  infinite  -  -  -  -  -  -     59 

44.  Determination  of  the  value  of  the  product  of  two  factors,  when  one  be- 

comes 0  and  the  other  ao-  -  -  -  -  -60 

45.  Determination  of  the  value  of  the  difference  of  two  functions,  when  they 

both  become  infinite  -  -  -  -  -  *  ib_ 

46.  Examples  on  the  preceding  theory  -  -  -  -  -  61 

47.  On  the  maxima  and  minima  values  of  functions  of  a  single  variable        -  63 

48.  If  the  function  F  (a  -j-  A)  be  developed  according  to  the  ascending  pow- 

ers of  h,  a  value  so  small  may  be  given  to  h  that  any  proposed  term 

in  the  series  shall  exceed  the  sum  of  all  that  follow  -  -    64 

49.  Determination  of  the  maxima  and  minima  values  in  those  cases  where 

Taylor's  theorem  is  applicable  -  -  -  -  -     ib. 

50.  Determination  of  the  values  when  Taylor's  theorem  is  not  appUcable     -    66 

.   .  dy 

51.  Maxima  and  minima  values  which  satisfy  the  condition  —  =  od  -    68 

ax 

52.  Conditions  of  maxima  and  minima,  when  the  function  is  impHcitly  given    ib. 

53.  Precepts  to  abridge  the  process  of  finding  maxima  and  minima  values         69 

54.  Examples  -  -  -  -  -  -  -  -70 

55.  On  the  cautions  to  be  observed  in  applying  the  analytical  theory  of  maxi- 

ma and  minima  to  Geometry  -  -  -  -  -    79 

56.  Differentiation  of  functions  of  two  independent  variables  -  -    81 

57.  Form  of  the  differential  when  the  function  is  implicit      -  -  -    82 


CONTENTS.  X? 

JkUele  Page 

SS.  The  ratio  of  the  two  partial  differential  coefficients  derived  from  u  =  Fz, 

z  being  a  function  of  a;  and  t/,  is  independent  of  F     -  -  -     83 

59.  Development  of  functions  of  two  independent  variables  -  -    84 

60.  The  partial  differential  coefficients  composing  the  coefficient  of  any  term 

in  the  general  development  are  identical  with  those  arising  from  dif- 
ferentiating the  preceding  term  -  -  -  -  -    87 

61.  Maclaurin's  theorem  extended  to  functions  of  two  independent  variables    88 

62.  Lagrange's  Theorem        -  -  -  -  -  -  -89 

63.  Applications  of  Lagrange's  theorem       -  -  -  -  -    91 

64.  Maxima  and  minima  values  of  functions  of  two  variables  -  -     94 

65.  Examples  -  -  -  -  -  -  -  -96 

66.  On  changing  the  independent  variable  -  -  -  -    99 

67.  On  the  failing  cases  of  Taylor's  Theorem  .  -  -  .  100 

68.  Explanation  of  the  cause  and  extent  of  these  failing  cases  -  -    ib, 

69.  Particular  examination  of  them  .  -  -  .  -  102 

70.  Inferences  from  this  examination  -----  103 

71.  The  converse  of  these  inferences  true     -----  104 

72.  To  obtain  the  true  development  when  Taylor's  theorem  fails     -  -    ib. 

73.  Correction  of  the  errors  of  some  analysts  with  respect  to  the  failing  cases 

of  Taylor's  theorem  ..----  106 

74.  On  the  multiple  values  of  —  in  implicit  functions        -  -  -  108 

dx 

75.  Determination  of  these  multiple  values  .  -  -  .  109 

<?« 

76.  Determination  of  the  multiple  values  of -p-r        -  -  -  -111 

ox* 


SECTION    II. 

Applicatiim  of  the  Differential  Calculus  to  the  Theory  of  Plane  Curves. 

77.  Explanation  of  the  method  of  tangents  -  -  -  -113 

78.  Equation  of  the  normal  -  -  -  -  -  -114 

79.  Apphcation  to  curves  related  to  rectangular  coordinates  -  -  116 

80.  Formulas  for  polar  curves  -  -  -  -  -  -117 

81.  Apphcation  to  spirals     .------  II9 

82.  Rectilinear  asymptotes  ------.  120 

83.  Examples  on  the  determination  of  asymptotes  -  -  -  122 

84.  Rectilinear  asymptotes  to  spirals  -  .  .  .  .  I23 

85.  Circular  asymptotes  to  spirals  .....  124 

86.  Expression  for  the  differential  of  an  arc  of  a  plane  curve  .  -  125 

87.  Principles  of  osculation  -  -  -  -  -  -126 

88.  Different  orders  of  contact         -  -  -  -  -  -  128 


XVI  CONTENTS. 

tSrticle  Page 

89.  Nature  of  the  contact  at  those  points  for  which  Taylor's  development 

holds  -  -  -  -  .  -  .  -  129 

90.  Of  the  contact  at  the  points  for  which  Taylor's  development  fails  -    ib. 

91.  Osculating  circle  -  -  -  -  -  -  -  130 

92.  Determination  of  the  radius  of  curvature  -  -  -  -    ib. 

93.  The  centres  of  touching  circles  all  on  the  normal  ...  132 

94.  Examples  on  the  radius  of  curvature      -  -  -  -  -    ib. 

95.  Expression  for  the  radius  of  cui-vature  of  an  ellipse  applied  to  determining 

the  ratio  of  the  polar  and  equatorial  diameters  of  the  earth     -  -  134 

96.  To  determine  those  points  in  a  given  curve,  at  which  the  osculating  circle 

shall  have  contact  of  the  third  order  ....  135 

97.  Expression  for  the  radius  of  curvature  when  the  independent  variable  is 

arbitrary  .....  -  -    ib. 

98.  Particular  fonns  derivable  from  this  general  expression  -  -  137 

99.  Suitable  formula  for  polar  curves  .....  138 

100.  Involutes  and  evolutcs  ......  I40 

101.  Determination  of  the  evolutcs  of  several  curves  -  -  -  141 

102.  Normals  to  the  involute  are  tangents  to  the  evolute       ...  143 

103.  The  difference  of  any  two  radii  of  curvature  is  equal  to  the  arc  of  the 

evolute  comprehended  between  them  -  .  -  -     ib 

104.  On  consecutive  lines  and  curves  .....  145 

105.  Determination  of  the  points  of  intersection  of  consecutive  curves  -    ib, 

106.  Determination  of  the  envelope  of  a  family  of  curves      ...  146 

107.  Examples  of  tliis  theory  ......  147 

108.  Multiple  points  of  curves  -  -  -  -  .  -150 

109.  Detennination  of  these  points  from  the  equation  of  the  curve   -  -  151 

110.  Conjugate  points  .......  J52 

111.  The  determination  of  these  point  does  not  depend  on  Taylor's  theorem    ib. 

112.  Multiple  points  of  the  second  species    -  .  -  .  .  153 

113.  Cusps  or  points  of  regression    -  -  -  -  -  -154 

114  Cusps  exist  only  at  points  for  which  Taylor's  theorem  fails      -  -    ib. 

115.  To  distinguish  a  limit  from  a  cusp         -  -  -  -  -    ib. 

116.  Examples  of  the  determination  of  cusps  whose  branches  touch  an  ordi- 

nate or  an  abscissa    -  -  -  -  -  .  -155 

117.  Cusps  whose  branches  touch  a  line  oblique  to  the  axes  -  -  156 

118.  Conditions  fulfilled  by  such  cusps  -  -  -  -      .       -  157 

119.  Distinction  between  cusps  of  the  first  and  those  of  the  second  kind      -    ib. 

120.  Examples         -  -  -  -  -  -  -  -ib. 

121.  On  points  of  inflexion   ----.-.  158 

122.  On  curvilinear  asymptotes        ......  igi 


CONTENTS.  XVll 

SECTION  III. 

On  the  general  Theory  of  Curve  Surfaces  and  of  Curves  of  Double  Curvature, 

tSrticle  Page 

123.  To  determine  the  equation  of  the  tangent  plane  at  any  point  on  a  curve 

surface  ........  164 

124.  Form  of  the  equation  when  the  equation  of  the  surface  is  implicit  -  165 

125.  To  determine  the  equation  of  the  normal  line  at  any  point  of  a  curve 

surface  -  -  -  -  -  -  -  -ib. 

126.  Expressions  for  the  inclinations  of  the  normal  to  the  axes         -  -  166 

127.  Forms  of  these  expressions  when  the  equation  of  the  surface  is  implicit     ib. 

128.  To  determine  the  equation  of  the  Unear  tangent  at  any  point  of  a  curve 

of  double  curvature    -  -  -  -  -  -  -     ib. 

129.  To  determine  the  eqtiation  of  the  normal  plane  at  any  point  of  a  curve 

of  double  curvature    -  -  -  -  -  -  -167 

130.  To  determine  tlie  equation  of  cylindrical  surfaces  in  general     -  -  168 

131.  General  differential  equation  of  cylindrical  surfaces      .  -  -  169 

132.  The  same  determined  otherwise  -  -  -  -  -     ib. 

133.  Given  the  equation  of  the  generatrix  to  determine  the  cylindrical  surface 

which  envelopes  a  given  curve  surface  -  -  -  -    ib. 

134.  If  the  enveloped  surface  be  of  the  second  order  the  curve  of  contact  will 

be  a  plane  curve  and  of  the  second  order         .  .  -  .  170 

135.  To  determine  the  general  equation  of  conical  surfaces  -  -    ib. 

136.  General  difierential  equation  of  conical  surfaces  ...  m 

137.  The  same  determined  otherwise  -  -  -  -  -    ib. 

138.  Given  the  position  of  the  vertex,  to  determine  the  equation  of  the  conical 

surface  that  envelopes  a  given  curve  surface  -  -  -  -     ib. 

139.  Mongers  proof  that  when  the  given  curve  surface  is  of  the  second  order 

the  curve  of  contact  is  a  plane  curve  ....  172 

140.  Shorter  method  of  proof  -  -  -  -  -  -    ib. 

141.  Davies^s  proof  that  there  is  one  point  and  only  one  from  which  as  a  ver- 

tex, if  tangent  cones  be  drawn  to  two  surfaces  of  the  second  order,  their 
planes  of  contact  shall  coincide  -  -  -  -  -  173 

142.  The  plane  through  the  curve  of  contact  is  always  conjugate  to  the  diame- 

ter through  the  vertex  of  the  cone       -  -  -  -  -  174 

143.  Surfaces  of  revolution  -  -  -  -  -  -  -    ib. 

144.T^o  determine  the  equation  of  surfaces  of  revolution  in  general  -    ib. 

145.  SimpUfied  form  of  the  equation  when  the  axis  of  revolution  coincides  with 

the  axis  of  z  --------  175 

146.  Remarkable  case,  in  which  the  generatrix  is  a  straight  line       -  -    ib. 

147.  General  differential  equation  of  surfaces  of  revolution  -  -  -  176 

148.  A  given  curve  surface  revolves  round  a  given  axis,  to  determine  the  sur- 

face which  touches  and  envelopes  the  moveable  surface  in  every  posi- 
tion     177 

C 


XVlll  CONTENTS. 

.article  Page 

149.  Example  in  the  case  of  the  spheroid      ....  -  178 

150.  Characteristic  property  of  developable  surfaces  ...  179 

151.  twisted  surfaces        -  -  -  -    ib. 

152.  Osculation  of  curve  surfaces    -  -  -  -  -  -    ib. 

153.  At  any  point  on  a  curve  surface  to  find  the  radius  of  curvature  of  a  nor- 

mal section    ..--.---  181 

154.  Elder's  theorem  viz.  at  every  point  on  a  curve  surface  the  sections  of 

greatest  and  least  curvature  are  always  perpendicular  to  each  other     182 

155.  Values  of  the  radii  of  curvature  of  any  perpendicular  normal  sections        ib. 

156.  Expressions  for  the  radii  of  greatest  and  ieast  curvatures  -  -    ib. 

157.  Peculiarities  of  the  surface  at  the  point  where  the  principal  radii  have 

different  signs  .......  183 

161.  Means  of  determining  when  the  signs  are  different       -  -  -  184 

162.  A  paraboloid  may  always  be  found  that  shall  have  at  its  vertex  the  same 

curvature  as  any  surface  whatever  at  a  given  point    -  -  -  185 

163.  To  determine  the  radius  of  curvature  at  any  point  in  an  obUque  section. 

The  theorem  ofMeusnier        -  -  -  -  -  -186 

164.  Lines  of  curvature        ...----  187 

165.  To  determine  the  lines  of  curvature  through  any  point  on  a  curve  sur- 

face  ib. 

166.  Lines  of  curvature  through  any  point  are  always  perpendicular  to  each 

other  ......--  189 

167.  On  the  developable  surfaces,  edges  of  regression,  &c.  generated  by  the 

normals  to  lines  of  curvature  .....  190 

168.  Radii  of  spherical  curvature     -.-..-  191 

169.  Given  the  coordinates  of  a  point  on  a  curve  surface  to  determine  the  ra. 

dii  of  spherical  curvature  at  that  point  -  -  -  -  192 

170.  The  radius  of  curvature  of  an  oblique  section  any  how  situated  with  re. 

spect  to  the  surface  and  to  the  axes  of  coordinates  is  now  determinable  ib. 

171.  To  determine  the  radii  of  curvature  at  any  point  in  a  paraboloid           -  193 

172.  Twisted  surfaces          -            -            -            -            -            -            -  194 

173.  To  determine  the  surfaces  generated  by  a  straight  line  moving  parallel 

to  a  fixed  plane  and  along  two  rectilinear  directrices  not  situated  in  one 
plane  ...--...  195 

174.  Two  straight  lines  shown  to  pass  through  every  point  on  the  surface  of 

a  hyperboUc  paraboloid  -  -  -  .  -  -196 

175.  To  determine  the  surface  generated  by  the  motion  of  a  straight  line  along 

three  others  fixed  in  position,  so  that  no  two  of  them  are  in  the  same 
plane 197 

176.  To  determine  the  surface  generated  when  the  directrices  are  not  all  pa. 

rallel  to  the  same  plane  -..-..  198 

177.  Two  straight  fines  shown  to  pass  through  every  point  on  the  surface  of 

ahyperboloid  of  a  single  sheet  -  -  -  -  -  199 

178.  On  twisted  surfaces  having  but  one  curvilinear  directrix  -  -  200 

179.  To  determine  the  general  equation  of  conoidal  surfaces  -  -    ib. 


CONTENTS.  WX 

^Hck  ^'^^ 

180.  Equation  of  the  right  conoid         -           -        -            -            -  -201 

181.  To  find  the  equation  of  the  inferior  surface  of  a  winding  staircase  -    ib. 

182.  To  determine  the  differential  equation  of  conoidal  surfaces       -  -  203 

183.  The  same  determined  otherwise           -            -            -            -  -    |b- 

184.  Twisted  surfaces  having  curvilinear  directrices  only     -            -  -    ib. 

185.  To  determine  the  general  equation  of  surfaces  generated  by  a  straight 

line  which  moves  along  any  two  dhectrices  whatever,  and  continues 
parallel  to  a  fixed  plane         ..----  204 

186.  Determination  ofthe  differential  equation  of  these  surfaces       -  -205 

187.  To  detenninc  the  general  equation  of  surfaces  generated  by  the  motion 

of  a  straight  line  along  tliree  curvilinear  directrices    -  -  -    ib. 

183.  Application  ofthe  preceding  theory     -  -  -  -  -207 

189.  Determination  ofthe  equations  of  the  intersections  of  consecutive  sur- 

faces   208 

190.  Determination  ofthe  general  equations  of  developable  surfaces  -  210 

191.  To  determine  the  developable  surface  generated  by  the  intersection  of 

normal  planes  at  every  point  in  a  curve  of  double  curvature  -  -  211 

192.  To  determine  the  developable  surface  which  touches  and  embraces  two 

given  curve  surfaces  -  -  -  -  -  -    ib. 

193.  To  determine  the  differential  equation  of  developable  surfaces  in  gene- 

ral      212 

194  The  same  determined  otherwise  .  .  .  -  -  213 

195.  Envelopes,  characteristics,  and  edges  of  regression      -  -  -    ib. 

196.  The  centre  of  a  sphere  of  given  radius  moves  along  a  given  plane  curve 

to  determine  the  surface  enveloping  the  sphere  in  every  position        -  215 

197.  On  curves  of  double  curvature  .....  216 

198.  Expression  for  the  differential  of  an  arc  of  double  curvature      -  -  217 

199.  Osculation  of  curves  of  double  curvature  -  -  -  -    ib. 

200.  Equation  ofthe  tangent  deduced  from  the  principles  of  osculation        -  218 

201.  To  determine  the  osculating  circle  at  any  point  in  a  curve  of  double  cur- 

vature -  -  -  -  -  -  -  -  219 

202.  General  expression  for  the  radius  of  absolute  curvature  -  -  221 

203.  Conditions  necessary  for  the  circle  to  have  contact  ofthe  first  order  only    ib. 

204.  Another  method  of  determining  the  osculating  plane    -  -  -    ib. 

205.  Another  method  of  determining  the  osculating  circle    -  -  -223 

206.  Other  and  more  simple  expressions  for  the  coordinates  of  the  centre  of 

tiie  osculating  circle  -  -  -  -  -  -  -ib, 

207.  To  determirte  the  centre  and  radius  of  spherical  curvature  at  any  point 

in  a  curve  of  double  curvature  .....  224 

208.  Determination  of  the  equations  of  the  edge  of  regression  of  the  developa- 

ble surfaces  generated  by  the  intersections  of  consecutive  normal 
planes  to  the  curve    -------  225 

209.  To  determine  the  points  of  inflexion  in  a  curve  of  double  curvature      -    ib. 

210.  On  the  evolutes  of  curves  of  double  curvature   -  -  -  -226 

211.  Lines  of  poles  -  -  -  -  -  -  -  -227 


3CX  CONTENTS. 

Article  Page 

212.  The  locus  of  the  poles  the  same  as  the  locus  of  the  characteristics       -    ib. 

213.  Every  curve  has  an  infinite  number  of  evolutes  all  situated  on  the  de- 

velopable surfaces  which  is  the  locus  of  the  poles       -  -  -    ib. 

214.  Every  curvilinear  evolute  of  a  plane  curve  is  a  helix  described  on  the 

surface  of  the  cylinder  which  is  the  locus  of  the  poles  of  the  plane  curve  228 

215.  The  shortest  distance  between  two  points  of  an  evolute  is  the  arc  of  that 

evolute  ........  229 

216.  Given  the  equations  of  a  curve  of  double  curvature  to  determine  those 

of  any  one  of  its  evolutes       -  -  -  -  -  -    ib. 

217.  To  prove  that  the  locus  of  all  the  linear  tangents  at  any  point  of  a  curve 

surface  is  necessarily  a  plane  -  -  .  .  .  231 

218.  Given  the  algebraical  equation  of  a  curve  surface  to  determine  whether 

or  not  the  surface  has  a  centre  -  .  .  _  .  232 

219.  To  determine  the  equation  of  the  diametral  plane  in  a  surface  of  the  se- 

cond order  which  will  be  conjugate  to  a  given  system  of  parallel  chords  234 

220.  A  straight  line  moves  so  that  three  given  points  in  it  constantly  rest  on 

the  same  three  rectangular  planes ;  required  the  surface  which  is  the 
locus  of  any  other  point  in  it  -  -  .  .  .  -235 

221.  To  determine  the  line  of  greatest  inclination    -  -  .  -  236 

222.  The  six  edges  of  any  irregular  tetraedron  are  opposed  two  by  two,  and 

the  nearest  distance  of  two  opposite  edges  is  caMed  breadth;  so  that 
the  tetraedron  has  three  breadths  and  four  heights.  It  is  required  to 
demonstrate  that  in  every  tetraedron  the  sum  of  the  reciprocals  of  the 
squares  of  the  breadths  is  equal  to  the  sum  of  the  reciprocals  of  the 
heights  ..---.-.  237 

Notes  .--.-...     249—265 


ERRATA. 
Page  33,  For  article  20,  read  article  23. 

183,  art  157,  at  bottom  of  page, /or  —  =  r',  read  —  =  tK 

r  a, 

184,  articles  158, 159, 160,  should  not  be  numbered. 


THE 


DIFFERENTIAL  CALCULUS. 


SECTION  I. 

05  THE 

DIFFERENTIATION  OF  FUNCTIONS  IN  GENERAL. 


CBAFTBR  Z. 

EXPLANATION  OF  FIRST  PRINCIPLES . 

Article  (1.)  All  quantities  which  enter  into  calculation,  may  be 
divided  into  two  principal  classes,  constant  quantities  and  variable 
quantities ;  the  former  class  comprehending  those  which  undergo  no 
change  of  value,  but  remain  the  same  throughout  the  investigation 
into  which  they  enter ;  while  those  quantities  which  have  no  fixed  or 
determinate  value,  constitute  the  latter  class. 

In  algebra  we  usually  employ  the  first  letters,  a,  b,  c,  &c.  of  the 
alphabet,  to  represent  known  quantities,  and  the  latter  letters,  z,  y,  x, 
&c,  as  symbols  of  the  unknown  quantities ;  but,  in  the  higher  calcu- 
lus, the  early  letters  are  adopted  as  the  symbols  of  constant  quantities, 
whether  they  be  known  or  unknown,  and  the  latter  letters  are  used  to 
represent  variables. 

Any  analytical  expression  composed  of  constants  and  variables,  is 
said  to  be  a.  function  of  the  variables.  Thus,  if  j/  +  ax^  -\-  bx  +  c, 
then  is  1/  a  function  of  x,  because  x  enters  into  the  expression  for  y ; 

1 


5S  THE  DIFFERENTIAL  CALCULUS. 

y  is  also  a  function  of  x  in  the  expressions  y  =  a"  -{-  b,  y  =  log.  x 
+  axr^,  &c.  and,  as  in  each  of  these  cases  the  fomi  of  the  function  is 
exhibited,  y  is  said  to  be  an  explicit  function  of  x ;  but,  in  such  equa- 
tions as 

axr  +  by^-\-cxy+  x  +  y  +  c  =  0,  x^  -{■  ary — aij-~y'^-\-  bx+  c,&c. 

where  the  form  of  the  function  that  y  is  of  x,  can  be  ascertained  only 
by  solving  the  equation,  y  is  an  implicit  function  of  x. 

Similar  remarks  apply  to  the  equations 

z  =  ax^  +  bif  +  ex  -i-  e  =  0,  az-  +  by^  +  cxz  +  e  =  0,  &c. 
2  being  an  explicit  function  of  x  and  y  in  the  first,  and  an  implicit 
function  of  the  same  variables  in  the  second  equation. 

If  we  wish  to  express  that  y  is  an  explicit  function  of  a:,  without 
writing  the  form  of  that  function,  we  adopt  the  notation  y  =  Fx,  or 
y  =  fx,  or  y  =  cpx,  &c.  and,  to  denote  an  implicit  function,  we  write 
F{x,y)  =  Oj{x,y)=0,&c* 

(2.)  Let  us  now  examine  the  effect  produced  on  the  function  y,  by 
a  change  taking  place  in  the  variable  x,  and,  for  a  first  example,  let 
us  take  the  equation  y  =  mx^.  Changing,  then,  x  into  x  -\-  h,  and 
representing  the  corresponding  value  of  y  by  y',  we  have 

y'  =  m  {x  +  hy 
or,  by  developing  the  second  number, 

y'  =  ma^  +  '2mxh  +  wi/i^. 

As  a  second  example,  let  us  take  the  equation  y  =  x^,  and  putting 
as  before  y'  for  the  value  of  the  function,  when  x  is  changed  into 
X  +  h,we  have 

y'  =    (^X+    hf  =  x''-\-  dxVl  +  Zxh?  +  h\ 

We  thus  see,  in  these  two  examples,  the  eflfect  produced  on  the 
function  by  changing  the  value  of  the  variable,  and,  on  account  of  this 
dependence  of  the  value  of  the  function  upon  that  of  the  variable,  the 
former,  that  is  y,  is  called  the  dependent  variable,  and  the  latter,  .r,  the 
independent  variable. 

*  In  this  general  mode  of  expression,  P,/,  and  <p,  are  mere  symbols,  represent- 
ing the  words  a  function  of:  thus,  Fx,  otfx,  means  a  function  of  a;,  thefonnofthe 
latter  differins'  from  that  of  the  former.  Ed. 


J 


THE  DIFFERENTIAL  CALCULUS.  3 

Let  us  now  ascertain  the  difference  of  the  values  of  each  of  the 
above  functions  of  x,  in  the  two  states  y  and  y'.     In  the  first  example, 

7/'  —  y  =  2mxh  +  mh?. 
In  the  second, 

y'  —  y  =  Sx'h  +  Sxh""  +  h\ 

so  that,  in  the  equation  y  =  map,  if  h  be  the  increment  of  the  variable 
X,  we  see  that  2mxh  +  mh^  will  be  the  corresponding  increment  of 
the  function  y ;  and,  in  the  equation  y  =  x^,  if  x  take  the  increment 
k,  the  corresponding  increment  of  the  function  will  be  Sstfh  +  3x/i^ 
+  li". 

We  may,  therefore,  in  each  of  these  cases,  readily  find  an  expres- 
sion for  the  ratio  of  the  increment  of  the  function  to  that  of  the  varia- 

y'  —  y 
ble,  that  is  to  say,  the  value  of  the  fraction  —- — -. 

In  the  first  case, 

•J^-^  =  2mx  +  h. 


In  the  second, 


h 


It  is  here  worthy  of  remark,  that  in  both  these  expressions  for  the 
ratio,  the  first  term  is  independent  of  h ;  so  that,  however  we  alter  the 
value  of  ^,  this  first  term  will  remain  unchanged.  If,  therefore,  h  be 
supposed  to  diminish  continually,  and,  at  length,  to  become  0,  the 
said  first  term  will  then  express  the  value  of  the  ratio.  This  first 
term,  then,  is  the  limit  to  which  the  ratio  approaches  as  h  diminishes, 
but  which  limit  it  cannot  attain  till  h  becomes  absolutely  0. 

In  the  first  of  the  foregoing  examples,  2mx  is  the  limit  of  the  ratio 

7/'  7/  .       .  .... 

■ — - — -  ;  or  it  is  the  value  towards  which  this  ratio  contmually  ap- 
proaches when  h  is  continually  diminished,  and  to  which  it  ultimately 
arrives  when  these  continual  diminutions  bring  it  at  length  to  h  =  0. 
In  the  second  example  the  limit  is  3x". 

(3.)  We  may  now  understand  what  is  meant  by  the  limit  of  the 
ritio  of  the  increment  of  the  function  to  that  of  the  variable.  It  is  the 
determination  of  this  limit,  in  every  possible  form  of  the  function,  that 
is  the  principal  object  of  the  diflbrcntial  calculus.     The  limit  itself  is 


«  THE  DIFFERENTIAL  CALCULUS. 

called  the  differential  coefficient,  derived  from  the  function ;  so  that, 
if  the  function  be  mi^,  the  differential  coefficient,  as  we  have  seen 
above,  is  2mx,  and  the  differential  coefficient  derived  from  the  func- 
tion, x%  is  3j:P. 

In  both  these  cases,  as  indeed  in  every  other,  the  respective  differ- 
ential coefficients  are  only  so  many  particular  values  of  the  general 

symbol  f ,  to  which  ^—  always  reduces  when  h  =  0.     In  the  first 

example  above,  f  =  2mx ;  in  the  second,  ^  =  3a;^. 

Instead  of  the  general  symbol  §,  a  particular  notation  is  employed 
to  represent  the  limiting  ratio,  or  differential  coefficient,  in  each 
particular  case ;  thus,  if  y  is  the  function,  and  x  the  independent  va- 

riable,  the  differential  coefficient  is  represented  thus,  -p.     If  z  were 
the  function,  and  y  the  independent  variable,  the  differential  coeffi- 
cient would  be  -r- ;  the  expressions  -~  and  -4-  have,  we  see,  the  ad- 
dy  ax  ay 

vantage  over  the  symbol  ^,  of  particularizing  the  function  and  the  in- 
dependent variable  under  consideration,  and  this,  it  must  be  remem- 
bered, is  all  that  distinguishes  ~  ox  -^  from  ^,  for  dy,  dz.,  dx,  are 

each  absolutely  0. 

This  notation  being  agreed  upon,  we  have,  when  y  =  mx^^ 


and,  when  y  =  aP, 


ax 
ax 


As  a  third  example,  let  the  function  y  =  a  -\-  3x-he  proposed, 
then,  changing  x  into  x  +  h  and  y  into  y',  we  have 
t/'  =  a  +  3x2  +  6xh  +  3/i', 

...  yjzl  =  ex  +  3h, 
n 

and,  making  fe  =  0,  we  have,  for  the  differential  coefficient, 

dx 
If  in  this  example  the  function  had  been  Sx^,  instead  of  o  +  3x ,  the 
differential  coefficient  would  obviously  have  been  the  same. 


THE  DIFFERENTIAL  CALCULUS*  5 

As  a  fourth  example,  let  y  =  aa^  db  b, 

.'.  y'  ~  ai?  dz  6  +  2axh  +  a/i^ 

.•.  y^IZl  =  •2ax  +  ah.'.-i^  =  2ax, 
h  dx 

which  would  have  been  the  same  if  the  constant  b  had  not  entered 

the  function. 

As  a  last  example,  take  the  function  y  —  (a  +  bxf  or  y  —  «^  + 

2abx  -j-  b^3^,  which,  when  x  is  changed  into  x  +  h,  becomes 

y'  =  0"  +  2ab  {x  +  h)  +  ¥{x-\-  hf 

=  a^  +  2abx  +  6^^  +  2  {ab  +  IPx)  h  +  b%% 

...  yjZlIl  =  2bia  +  bx)  +  b% 
h 

...  iL  =  2b(a  +  bx). 
dx 

It  should  be  remarked,  that  of  the  two  parts  dy,  dx,  of  which  the 

dy 
symbol  -~  consists,  the  former  is  called  the  differential  o£y,  and  the 

latter  the  differential  of  x.  These  differentials,  although  each  ==  0, 
have,  nevertheless,  as  we  have  already  seen,  a  determinate  relation 
to  each  other ;  thus,  in  the  last  example,  this  relation  is  such,  that 
di/  =  26  (a  4"  bx)  dx,  and,  although  this  is  the  same  as  saying  that 
0  =  26  (o  +  bx)  X  0 ;  yet,  as  we  can  always  immediately  obtain 

dii 
from  this  form  the  true  value  of  ^  or  -p,  we  do  not  hesitate  occa- 
sionally to  make  use  of  it. 

From  the  expression  for  the  differential  of  a  function,  we  readily 

see  the  propriety  of  calling  ^  a  coefficient,  being,  indeed,  the  coeffi- 
cient  of  da;. 


OBAPTSn  zx. 


DIFFERENTIATION  OF  FUNCTIONS  OF  ONE 
VARIABLE. 

(4.)  Let/c  represent  any  function  of  x  whatever,  then,  if  x  be  chan- 


y  THE  DIFFERENTIAL  CALCULUS. 

ged  into  x  +  h,  the  general  form  of  the  development  of/ (a:  +  /i), 
arranged  according  to  the  powers  of /i,  will  be 

f{x-\-  h)  =-fx+  Ah  +  B/r  +  Civ'  +  D/i^  +  &c. 
as  may  be  proved  as  follows  : 

;  1.  The  first  term  of  the  development  must  be  fx.  This  is  obvious, 
for  this  first  term  is  what  the  whole  development  reduces  to  when 
h  ■=  0,  but  we  must  in  this  case  have  the  identity /r  =/c;  hence, 
fx  is  the  first  term. 

2.  JVbne  of  the  exponents  of  h  can  he  fractional.  For  if  any  expo- 
nent were  fractional,  the  term  into  which  it  enters  would  be  irrational, 
so  that  the  development  oif(x  +  h),  containing  an  irrational  term, 

f{x  +  h)  itself  would  be  irrational,  since  it  is  impossible  that  there 
should  be  an  equality  between  an  irrational  expression  and  one  that 
is  rational.  But,  if /(a?  +  h)  were  in'ational,/c  would  likewise  be 
irrational,  for  the  former  function  differs  from  the  latter  only  in  this, 
that  x  +  h  occupies  in  it  the  place  of  x  in  the  latter ;  and,  therefore, 
as  many  values  as  fx  has,  in  consequence  of  the  radicals  that  may 
enter  into  it,  so  many  values,  and  no  more,  must/(a;  +  h)  have.  As, 
therefore,  the  first  term  of  the  development  off{x  +  h)  has  the  same 
number  of  values  a.sf{x  +  h)  itself,  none  of  the  succeeding  terms 
can  contain  a  fractional  power  of  h ;  for,  otherwise,  the  development 
would  have  more  values  than  its  equivalent  function,  which  is  absurd. 

3.  JYone  of  the  exponents  of  h  can  be  negative  in  the  general  de- 
velopment. For,  on  the  supposition  that  a  negative  power  of  h  en- 
ters any  term,  that  term  would  become  infinite  when  /i  =  0 ;  but, 
when  h  =  0,  the  function  is  simply /c,  which  is  not  necessarily  infi- 
nite ;  so  that,  that  development  into  which  a  negative  power  of  h  en- 
ters, cannot  be  the  general  development  o(f(x  +  h),  understanding, 
by  the  general  development,  that  which  does  not  restrict  x  to  any  par- 
ticular value  or  values.  By  supposing  a  negative  power  of /i  to  ap- 
pear in  the  development,  we  have  just  seen  that  such  development 
would  restrict  x  to  the  values  determined,  by  the  equation  fx  =  cc, 

or  by  the  equation  -r^  =  0,  to  the  exclusion  of  all  other  values,  and 

is,  therefore,  not  general.* 

*  It  has  been  shown,  that  the  general  development  of  fx,  when  a;  becomes 
X  -\-  h,  is  f{x  +  h)  =fx-{-  All"-  +  B/i*  -|-  C/t"^  +  &c.,  in  which  the  exponents 
c,  b,  c,  Sac.  of  the  increment  h  of  tlie  variable,  are  whole  and  positive  numbers,  it 


THE    DIFFERENTIAL    CALCULUS.  7 

As  to  the  method  of  determining  the  coefRcients  A,  B,  C,  &c.  that 
will  be  investigated  hereafter  {in  Chapter  iv.) 

4.  It  must  be  here  remarked,  that,  although  the  general  develop- 
ment of  every  functiou  of  x  is  of  the  above  form,  yet  we  are  not  to 
suppose  that  this  form  will  remain  unchanged  whatever  particular 
value  we  give  to  x,  for  particular  values  may  be  so  chosett  as  to  ren- 
der this  form  of  development  impossible,  and  such  impossibility  will 
be  intimated  by  the  assumed  value  of  .r  rendering  some  of  the  coeffi- 
cients A,  B,  C,  &c.  infinite.  The  well  known  binomial  theorem, 
which  we  already  know  to  be  of  the  above  form,  will  affond  an  illus- 

yet  remains  to  prove,  tlmt  they  will  be  represented  by  the  series  1, 2,  3,  &c.  as  as- 
sumed in  the  commencement  of  tliis  chapter. 

It  is  already  known,  that  the  first  term  of  the  development  of /(x  +  ft),  is  the 
primitive  function, /a;,  and  that  the  remainder  must  disappear  when  A  =  0,  which 
remainder  must  then  contain  A  as  a  factor,  so  that  we  shall  have 

f{x  +  h)=fx  +  Vh (1), 

and  thence 

h 
P  being  a  new  function  of  a;  and  h,  can  likevrise  be  developed,  of  which  the  first 
term  will  be  the  value  P  will  assume  when  A  =  0,  which  we  will  represent  by  p, 
and  as  the  remaining  terms  must  vanish  when  A  =  0,  they  must  contain  A  as  a 
factor ;  we  thus  have 

P  =  p  +  ah, 
which  substituted  in  equation  (1),  gives 

fix  +  h)=fx  +  ph  +  OLh^,  &c (2) 

again  we  have 

-= V^ 

in  which  Q,  is  another  function  of  a:  and  h,  and  may  be  developed  the  same  as  P ; 
the  first  term  of  this  development  will  be  the  value  of  d  when  A  =  0,  which  we 
represent  by  q,  and  the  remainder  becoming  zero  when  A  =  0,  must  contain  A  as 
a  factor ;  we  shall  then  have 

a  =  g  -f-  RA, 
in  which  R  is  another  function  of  a;  and  A.    Expression  (2),  will  thus  become 

f{x  -\-  A)  =fx  +  ph  -f  qh^  -I-  RA3  +  &c. 
being  the  general  form  of  the  development  of /(x  -j-  A),  in  which/),  q,  r,  &c.  arc 
functions  of  a;  alone,  and  correspond  to  the  coefficients  A,  B,  C,  &c.  in  the  article 
under  consideration.  Ed. 


8  THE    DIFFERENTIAL   CALCULUS. 


tration  of  this.     Thus  the  general  development  of/c  =  >/  x  -\-  a, 
when  we  replace  a:  by  a:  +  h,  is,  by  the  above  mentioned  theorem, 


f\{x-\-  h)  -{-  a\  =  V{x-\-a)  +  h  = 

{x  +  «)^  +  i{x  +  a)-^  h—  :^{x  +  af^  h^  +  &c. 
where,  in  the  case  x  =^  —  a,  all  the  coefficients  become  infinite,  and 
the  develqiment,  according  to  the  positive  integral  powers  of  ^-,  be- 
comes in  this  case  impossible ;  for  the  function  then  becomes  merely 
\/  h  or  h\,  in  which  the  exponent  of  ^  is  fractional.  The  impossibility 
of  the  proposed  form  of  development  in  such  particular  case  is  always 
intimated,  as  in  the  example  just  adduced,  by  the  circumstance 
of  infinite  coefficients  entering  it,  for  imaginary  coefficients  would 
imply  merely  that  the  function  f{x  +  h)  for  the  assumed  value  of  a; 
becomes  imaginary,  and  not  that  the  development  failed.  A  particu- 
ar  examination  of  the  cases  in  which  the  general  form  of  the  deve- 
lopment fails  to  have  place,  will  form  the  subject  of  a  future  chapter ; 
at  present  it  is  sufficient  to  apprise  the  student  that  such  failing  cases 
may  exist. 

(6.)  By  transposing  the  first  term  in  the  general  development  of 
f{x  +  ^.),  we  have 

f{x  +  h)  —fx  =  A^  +  Bli"  +  C/i='  +  &c. 

,.  n^±^-Z±.  =  A  +  Bh+  Ch^  +  &c. 
h 

hence,  when  h  =  0, 

dx 
from  which  result  we  learn,  that  the  coefficient  of  the  second  term,  in 
the  development  of  the  function  f{x  +  h),  is  the  differential  coefficient 
derived  from  the  function  fx ;  so  that  the  finding  the  difierential  coef- 
ficient from  any  proposed  function,  fx,  reduces  itself  to  the  finding 
the  coefficient  of  the  second  term  in  the  general  development  of 
f{x  +  h),  or  of  the  first  term  in  the  developed  difference  f{x  +  h) 

-fa- 

Having  obtained  this  general  result,  we  may  now  proceed  to  apply 

it  to  fimctions  of  different  terms ;  but  it  will  be  proper  previously  to 

observe,  that  those  constants  which  are  connected  with  the  variable 

in  the  functiouyx,  only  by  way  of  addition  or  subtraction,  cannot  appear 

in  the  coefficient  A  ;  because  A,  being  multiplied  by  h,  can  contain 


THE  DIFFEHENTIAL  CALCULUS.  9 

no  quantities  which  are  not  among  those  multiplied  by  x  +  ^  in 
f{x  -\-  h),  or  by  x  infx. 

(6.)  To  differentiate  the  product  of  two  or  more  functions  of  the 
same  variable. 

Let  y,  2,  be  functions  of  x,  in  the  expression 
u  =  ayz. 
By  changing  x  into  x  +  h,  the  function  y  becomes 

1/  =y+  Ah+  Bk"  +  Ch^  +  &c.  .  .  .  (1), 
«nd  the  function  z  becomes 

z'  =  z-\-  A7i  +  B7i2  +  CV-{-  &c.  .  .  .  (2). 
Hence,  when  ^  =  0,  we  have  from  (1) 

h  da?         ' 

and  from  (2) 

z'  —  z  _  dz  _  ., 
h  dx      '  ' 

Muhiplying  the  product  of  (1)  and  (2)  by  a,  we  have 
u'  =  ayz  -f  a  {Az  +  A'y)  h  +  &c.* 

=  ayz+a{^^z+^y)h  +  kc. 

therefore,  af-pz4-  -r-y)  being  the  coefficient  of  the  second  term 

of  the  development  of  «',  we  have 

du  dy   ,        dz 

-J- =  az -~ -t  ay  -;- 
dx  dx  dx 

.'.  du  =■  azdy  +  aydz  .  .  .  (3). 
Hence,  to  differentiate  the  product  of  two  functions  of  the  same  va- 
rictble,  we  must  multiply  each  by  the  differential  of  the  other,  and  add 
the  results. 

It  will  be  easy  now  to  express  the  differential  of  a  product  of  three 
functions  of  the  same  variable.     Let. 

u  =  wyz 
be  the  product  of  three  functions  of  .r ;  then,  putting  v  for  wy,  the  ex- 
pression is 

u  =  vz; 
hence,  by  (3), 

*  tt'  is  that  value  wliich  w  attains  when  the  functions  y  and  z  have  varied  by 
virtue  of  the  variation  h  of  the  variable  x  on  which  they  depend.  Ed. 

2 


10  THE  DIFFERENTIAL  CALCULUS. 

du  =  zdv  +  i^dz, 
butt)  =  toy ;  therefore,  by  (3),  dv  =  ydic  +  wyd ;  consequentiy, by 
substitution, 

dn  =  sydw  +  ziody  +  wydz  .  .  .  (4), 
and  it  is  plain  that  in  this  way  the  differential  may  be  found,  be  the 
factors  ever  so  many ;  so  that,  generally,  to  differentiate  a  product  of 
several  functions  of  the  same  variable,  ice  must  multiply  the  differen- 
tial of  each  factor  by  the  product  of  all  the  other  factors,  and  add  the 
results. 

If  we  suppose  the  factors  to  be  all  equal  to  each  other,  we  shall 
obtain  a  rule  to  differentiate  a  positive  integral  power.  Thus  the 
differential  of  the  function 

U  =  X^  =  X'X'X'X.... 

is 

du  =  af*-'  dx  +  af^*  dx  +  af"'^  dx  +  &c.  to  m  terms, 
that  is 

du  =  maf^^  dx  .•.  —-  =  m3f^\ 
ax 

This  form  of  the  differential  is  preserved  whether  m  be  integral  or 
fractional,  positive  or  negative ;  but,  to  prove  this,  we  must  first  dif- 
ferentiate a  fraction. 

(7.)  To  differentiate  a  fraction.    Let  u  =  -,y  and  z  being  func- 
tions of  X ;  therefore  uz  =  y,  and  duz  =  dy,  that  is,  by  the  last  article, 
zdtc  +  udz  =  dy  .:  du  =  — , 

or,  substitutmg  -  for  u, 

^^^zdy-ydz^ 
z^ 
Hence,  to  differentiate  a  fraction,  the  rule  is  this  :  From  the  product 
of  the  denominator,  and  differential  of  the  numerator,  subtract  the 
product  of  the  numerator,  and  differential  of  the  denominator,  and 
divide  the  remainder  by  the  square  of  the  denominator. 
(8.)  To  differentiate  any  power  of  a  function. 
The  form  of  the  differential  when  the  power  is  whole  and  positive 
has  been  already  established.     Let  then 


THE  DIFFERENTIAL  CALCULUS.  11 

m 

u  =  y" 

be  proposed,  y  being  a  function  of  a?,  and^  being  a  positive  fraction. 
Since  u"  =  tj'", 

.'.  nw""'  du  =  my"^^  dy, 

Now 


consequently, 


, .        mil  —  m        m 

(m  ~  1) = 1, 

n  n 


du  =  —  y        dy. 


Let  now  the  exponent  be  negative,  or 

u  =  y-H 

1 

.*.  m"  =  v~"'  =  — 

J  yrn 

.*■.  du"  =  d  — 
but 

.'.  nW^^  du  =  —  jn?/~""~'  dy, 
and  dtt  = — -—  dy, 

or,  substituting  for  u  its  equal  y~'^,  we  have 


Jn"  =  nM^■'d«,  and  d  -;j  =  —  m  -^-^;j-  dy  =  —  wj/"*""'  dy, 


m 


du  = y  dy. 

Hence,  generally,  to  differentiate  a  power,  ive  must  multiply  together 
these  three  factors,  viz.  the  index  of  the  power,  the  power  itself  dimi- 
nished  by  unity,  and  the  diffei'ential  of  the  root. 

This  rule  might  have  been  deduced  with  less  trouble,  by  availing 
ourselves  of  the  binomial  theorem,  for,  supposing  inu  =  y^  that  the 
increment  of  the  function  y  becomes  k  when  the  increment  of  a?  be- 
comes h,  we  have  u'  =  {y  +  ky  and,  by  the  binomial  theorem,  the 


12  THE  DIFFERENTIAL  CALCULUS. 

coefficient  of  the  second  term  of  the  expansion  of  {y  +  ky  is  py''~\ 
whether  p  be  positive  or  negative,  whole  or  fractional.  As,  however, 
we  propose  to  demonstrate  the  binomial  theorem  by  means  of  the 
differential  calculus,  we  have  thought  it  necessary  to  establish  the 
fundamental  principles  of  differentiation,  independently  of  this  theo- 
rem. 

(9.)  If  it  be  required  to  differentiate  an  expression  consisting  of 
several  functions  of  the  same  variable,  combined  by  addition  or  sub- 
traction, it  will  be  necessary  merely  to  differentiate  each  separately, 
and  to  connect  together  the  result?  by  their  respective  signs.  For 
let  the  expression  be 

M  =  ato  +  fcy  +  c2  +  &c. 
in  which  w,  y,  c,  are  functions  of  x.     Then,  changing  x  into  x  -{■  h 
and  developing, 

to  becomes  w  +  Ah    +  B^^    +  &c. 

y  y  +  A'h+  B'h'  +  &c. 

s  2  +  A"h  +  B"h'  +  &c. 

.-.  «  M  +  (aA  +  6A' +  cA"  +  &c.) /i  +  &c. 

.*.  du  =  aAdx  +  bA'dx  -\-  cA"dx  +  &c. 

But 

Adx  =  dWf  A'dx  =  dy,  A"dx  =  dz,  &c. 
therefore 

du  =  adw  +  bdy  +  cdz  +  &c. 
that  is,  the  differential  of  the  sum  of  any  number  of  functions  is  equal 
to  the  sum  of  their  respective  differentials. 

(10.)  We  shall  now  apply  the  foregoing  general  rules  to  some 
examples. 

EXAMPLES. 

1.  Let  it  be  required  to  differentiate  the  function 
y  =  8x*  —  3aP—  5x. 
By  the  rule  for  powers  (8)  the  differential  of  3x*  is  8  X  4x^dx,  and 
the  differential  of  —  SxP  is  —  3  X  Sx^dx ;  also  the  differential  of 
—  5x  is  —  5dx  ;  hence  (9), 

dy  =  32x'dx  —  9ordx  —  5dx, 

...  $  =  32ar-'— 9x2  —  5. 
dx 


THE  DIFFERENTIAL  CALCULUS.  13 

2.  Let  t/  =  (a:"'  +  a)  {3x^  +  b). 
By  the  rule  for  differentiating  a  product  (6),  we  have 

dy  =  {aP  +  a)d  (3r'  +  5)  +  (St"  +  &)  d  (r*  +  a), 
and  (8), 

d  {Sr'  +  6)  =  6xdx,  d  [aP -{■  a)  =  Sr'dx, 
.-.  dy  —  {3p  +  a)  exdx  +  (Sx^  +  6)  SsPdx, 

ax 
3.  Let  1/  =  (ax  +  a;^)^. 

The  differential  of  the  root  ox  +  ar  of  this  power,  is  orfx  +  2xdxt 
therefore, 

dy  =  2  {ax  +  x^)  {a  -{-  2x)  dx, 

.'.■£  =2  {ax  +  x")  {a-\-  2x). 


4.  het  y  =  y/a-^bx~. 

The  differential  of  the  root  or  function  under  the  radical,  is  2bxdx ; 
hence 

1  f)x 

%  =  1  («  +  bx')-^  2bxdx  =  — dx, 

>/a+  bxP 

dy bx 

'  '  dx       ^a-^-bif 

5.  Let  y  =  {a  +  baf")". 

The  differential  of  the  root  or  function  within  the  parenthesis,  is 
mbaf^^dx;  hence 

dt/  =  n  (o  +  6x")"-'  m6a?'"~'  dx, 

...  -1  =  imn  (a  +  6*'")""'  **""'• 
ax 

x" 

6.  Let  j(  = 


(a  +  x3)2 

The  differential  of  the  numerator  of  this  fraction  is  2xdx,  and  the 
differential  of  a  +  x'  is  3x^dx,  therefore  the  differential  of  the  de- 
nominator is  2  (a  +  x^)  3x^dx ;  hence  (7), 

,    _{a+  apy  2xdx  —  6x^  (a  +  x^)dx  _  2ax~4x^ 
^  (a  +  x^y  ~  (a  +  aPy     '"' 

dy  _  2x{a  —  2aP) 
'''  dx         (a  +  apy    ' 


14  THE  DIFFERENTIAL  CALCULUS. 


7.  Let  7/=  \a+  ,/(6  +  ^j^ 
The  differential  of  the  root  a  +  \/(6  +  ^)  is  ^  (6  +  ^)  ~* 

d  -5-,  and  d-^  = 7-  cb ;  hence 

ST  or  or 


^  >/b+^ 


8.  Let  y=Var'  +  Va+x'. 
The  differential  of  ic^  +  \/a  +  a^  is  2xdx  +  (a  +  ar')-2  ardar, 

djf  X  X 

...  ^  =  =  +         " 


9.  Lety  =  — . 

Va^  +  x^  —  X 


Multiplying  numerator  and  denominator  by  -v/a^  +  ^  +  x,  the 
expression  becomes 

"^        a         a 


.-.  %  =  d  ^  +  ^^ — ^ ^^+^  d  VaP  +  a^. 


ar^ 


dy  _  2a:    ,    Va^  +  ar^    I 
•'•die  ~  ^  a^  ^2  ^ a=»  +  ar' 

2a:  a"  +  2a^ 

=  TT  + 


a^"  Va"  +  x" 

10.  y  =  a?—.x'.:-^  =  —  2x. 

11.  u  =  4ar'  — 2ar'  +  7x  +  3  .-.  -j^  =  12ar'  — 4a:+  7. 

ax 


^ 


THE    DIFFERENTIAL    CALCULUS.  15 

12.  y={a+  hx)  x" .'.  J  =  46^*  +  Sax^. 

13.  y  =  {a -^  bx -\-  ex"  +  &c.)'".-.  ^  =  w  (a  +  6x  + car* 
+  &c.)'^'  {b-\-cx-\-  &c.) 

14.  y  =  (a  +  6r^)^.-.^  =  ^y^+6^. 

I*;       =       ,.    4yar         dy  _    6(1— ar') 
!»•  2^       a  -f-  3  ^  ^  •  •  ^^       (3  +  a^)2  v/x 

,    ,     ,         c      dy  b       ,     c 

17.  y  =  (ox'  +  6)'  +  is/cf—e  (x—h)  .:  ^  =  6111? 


v/a2 


18.  y  = 


dy  _ 


a:+  y/l—r"  '  '  dx         Vl—x'{l-\-2xy/l-r') 
The  functions  in  these  examples  are  all  algebraic,  we  shall  now 
consider 

Transcendental  Functions. 
(11.)  Transcendental  functions  are  those  in  which  the  variable 
enters  in  the  form  of  an  exponent,  a  logarithm,  a  sine,  &c.  Thus, 
a',  a  log.  X,  sin.  x,  &c.  are  transcendental  functions :  the  first  is  an 
exponential  function,  the  second  a  logarithmic  function,  and  the  third 
a  circular  function. 

To  find  the  Differential  of  a  Logarithm.* 

(12.)  Let  it  be  required  to  differentiate  log.  x. 

Put  a  for  the  base  of  the  system  of  logarithms  used,  and  let 

M  = 1 1 

a_l_i(a_l)2  +  i(a_l)3_&c.' 

then 

log.  (1  +  n)  =  M  (n  —  i  n=  +  i  »^  —  &c.) 

or,  putting  -  for  », 

X 

*  Note  (A'). 

t  Algebra,  Chap.  vii.  p.  219.,  or  vol.  i.  p.  155,  Lacroix's  large  work  on  the  Dif- 
ferential Calculus.  Ed. 


16  THB    DIFFERENTIAL    CALCULUS. 

.log.(.+  fe)-log..^^l_    fe  j^_ 

This  is  the  general  expression  for  the  ratio  of  the  increment  of  the 
function  to  that  of  the  variable.  Hence,  taking  the  limit  of  this  ratio, 
we  have 

d  log.  X  _  M  ^^^ 
dx  X  '  '  ' 

If  the  logarithms  employed  be  hyperbolic  M  =  1,  and  then 

d\og.x_  1 
dx  X  '  ' 

If  they  are  not  hyperbolic,  write  Log.  instead  of  log.  for  distinction 
sake,  then,  since  by  putting  a  for  1  +  » in  the  series  for  log.  (1  +  n), 
we  have 

log.  a  =  a-  1  -1  (a-  If  +  1  {a-lY  —  &c.=~ 

it  follows,  from  the  expression  (1),  that 
d  Log.  X  _        1 


dx  log.  a  .  X 

Unless  the  contrary  is  expressed,  the  differential  is  always  taken  ac- 
cording to  the  hyperbolic  system,  because  the  expression  is  then 
simpler,  log.  a  being  =  1. 

From  the  preceding  investigation  we  learn,  that  the  differential  of 
a  logarithmic  function  is  equal  to  the  differential  of  the  function  di- 
vided by  the  function  itself. 

(13.)    To  differentiate  an  exponential  function. 

1.  Let  y  =  (f  then  log.  y  =  x  log.  a  .•.  d  log.  y  =  dxlog.  a,  that 

is,  —  =  dx  log  a  .•.  dy  ="  y  log.  a.dx  =  log.  a  .  a'  dx. 

Hence,  to  differentiate  an  exponential^  we  mu^t  multiply  together  the 
hyp.  log.  of  the  base,  the  exponential  itself,  and  the  differential  of  the 
variable  exponent. 


EXAMPLES. 


1.  Let  y  =  X  {a:''  +  xr)  V  d'  —  x^ .-.  log.  y  =  log.  x  +  log. 
(o'  +  x^)  +  i  log.  (a^  --  r-), 


THE    DIFFERENTIAL   CALCULUS.  17 

dy  _  dx         2xdx  xdx     _      a^  +  aV  —  4x*       , 

'  '   y         X        a^  +  x^       c?  —  X'       x{o?  -\r  x^)  {a?  —  x") 
therefore,  substituting  for  ij  its  value,  we  have, 
dy   _   «"  +  «'-<^  —  4a;^ 
dx  y/aj^  —  x^ 


\/  a  +  a?  +  \/«  —  x 
2.  y  =  log.  -== ;-==.• 

Multiplying  numerator  and  denominator  by  the  denominator,  the 
expression  becomes 

2x 


y  =  log. ==  =  log.  X  —  log.  (a  —  \/  a^  —  x^) 

2a  —  2  \/  a^  —  x^ 

dy  _    1  X  _       a  y/d^  —  x^  —  a^ 

^^         ^        a  VaF  —  or  —  a^  +  ^^         x\/a? — x^\a — V  d^ — x^\ 

—  a 


X  \f  a^  —  a^ 


\/  x^  -\-  2ax 
3.  y  =  r7=F====^  •••  log.  y  =i  log.  {x"  +  2ax)  —  i 
^xr  -\-  x-  —  X 

log.  {3p-\-a^  —  x)^ 

dy  X  +  a      ,  3x^  -{-  2x  —  1      , 

.♦.-^  —  — ; dx  —  „  ^  .,    ■ — dx  = 

y         or  +  2ax  3  {x   +  x"  —  x) 

(l-^Sa)  x'~{a  +  2)  x  —  a 

3x{x'+  x—l)  {x  4-  2fi.)  ' 

.  dy  _\{1  —  3a)  x^—{a  +  2)  x  —  a\  V  x 

^^  Sx'^  (x'  +  a;  —  1)  3  {x  +  2a)"3' 


dy 


4.  J/  =  x""^"  * .'.  log.  y  =  m  ^  —  1  log.  X  .'.~  =  m  ->/  —  1 
dy  y 


dx 


X 


.  .  -z-  =m~  V  —  1  —  m  \/ i.x"^-i-i. 

dx  X 

From  </j,zs  example  it  appears,  that  the  rule  at  (8)  applies  when  the 
exponent  is  imaginary. 

6.  t/  =  a'  . 
In  this  example  the  variable  exponent  is  x* ;  hence,  calling  it  z  and 
taking  the  logarithms,  we  have 

3 


18  THE    WFFEKENTIAL    CALCULrS. 

dz  xdor 

log.  z  =  X  log.  X  .'.  —  —  log.  xdx  +  — '- .'.  dz  =  3f{\  +log.ar)f^5 

Z  X 

hence,  by  the  rule, 

-J-  =  log.  a  .  a*'  .  af  (1  +  log.  x). 
ax 

6.  y  r=:e»v'-'-j-e-»>/-',  where  c  is  the  base  of  the  hyper- 
bolic system, 

.-.  dy  =  c'>/~   v/  — 1  dx  —  e-> ^~^  v/  —  1  dar, 

7.  y  =  log.  (log.  x).*  Put  2  for  log.  a?.-,  y  =  log.  «.•.  dy  =  — 

but  dz  =  d  log.  X  ==—.'.  dy  =  — r-^ —  •••  t^  =  — ; • 

X  "^        X  log.  ar       ax       X  log.  x 

o  „  .         dy       mn  (log.  x")*^* 

9.  y  =  log.  J  (a  +  X)'  (a  +  x)-  (a"  +  x)-"S  •••  ^  =  J^ 

j^      m'  m" 

a    -\-   X       a    -f-  X 

,„  ,        Va  +  \/x       dy  y/a 

10.  y  =  log. .*.  — ^  = . 

^  Va  —  Vx      dx       (a  —  x)  Vx 

11.  t/  =  c^...  ^  =  6**  .  x*  (1  +  log.  x). 

12.  y  =  (log.)''x.-.^  = 


dx       X  log.  X  (log.)"  X  .  .  .  (log.)""'  X. 


,-    ,  v/  1  +  x^       dy  1 

13.  log.  y  = ■ .'.  -i  = -. 

^  ^  X  dx  ar^ 

14.  y  =  a*',  2  being  a  function  of  X,  .*.  ~  =  log.olog.&.a*'6^-v- 
*  This  means  tho  logarithm  of  the  logarithm  of  x,  but  the  notation  we  shall 

hereafter  adopt  will  be  (log.) 'a;,  and  which  we  shall  extend  to  circular  functions ; 
thus,  instead  of  sin.  (sin.  x),  we  shall  write  (sin.)^a;,  the  square  of  the  sine  being 
written  without  the  parenthesis,  thus,  sin.  %.  We  may  call  such  expressions  a« 
(log.)"  X,  (sin.)"  X,  &.C  the  nth  log.  of  i,  the  nth  sine  of  j:,  &c 


■ 


THE  DIFFEaENTIAL  CALCULUS.  19 

15.  y  =  a*'""*"  .-.  ^  =  log.  o  .  log.  b  .  a*'''+'  .  h''  +'(2^+1) 


dx 

ly 

dx 


du       log.  a  .  a  ^°^^ ' 
16.  y  =  a  '"s*  .     J  —     & 


,     n         dy  e  <'°«>    ' 

17.  y  =  e  ''"^-^  *  .'.  —  = . 

'  ^  '  '  dx       log.  a;  (log.)^a^  ....  (log.)""'  x 

18.  y  =  x''  .'.^=  af'.af\l-\-  log.  x  (1  +  log.  x)|. 

(14.)   To  differentiate  circular  functions. 

Let  x  represent  the  versed  sine  of  an  arc  of  a  circle  whose  radius 
is  r,  then  r  —  x  will  represent  the  cosine  of  the  same  arc,  and,  by 
trigonometry, 

tan.  _      r 
sin         r  —  x' 
In  this  expression,  x  is  the  independent  variable,  and  as  this  dimin- 
ishes, the  arc  itself  diminishes,  both  vanishing  simultaneously,  and  the 

tan.  .    r 
ultimate  ratio  of  — r-  is  -  =  1 ;  that  is,  the  sine  and  tangent  of  an  arc 
sm.      r 

approximate  to  each  other  as  the  arc  diminishes,  and  at  length  become 
equal.     As  the  arc  is  between  the  sine  and  tangent  when  these  be- 
come equal,  the  arc,  also,  must  become  equal  to  each ;  therefore,  we 
may  conclude,  that  the  ultimate  ratios  are  as  follows  : 
tan.  _       arc  _      arc  _        arc    _        sin.    _        tan.    _      ^ 
sin.         '  sin.         '  tan.         '  chord         '  chord         '  chord 

1.  Let  it  now  be  required  to  find  the  differential  of  sin.  x.  Chang- 
ing X  into  a;  +  fc,  we  have  {Gregorxfs  Trig.  p.  48) 

sin.  (x  +  ft.)  =  sin.  a;  +  2  sin.  \  h  cos.  {x  -\-  \  A), 

sin.  ix  -\-  h)  —  sin.  x       sin.  \h  ,      ■    ,  ,  v 

•••  — ^ — r — -  =  -it  ■=""  '^  +  i  *" 

sin.  ^h       ,        d  sin.  x 

whenx=  0, ■--  =  1,  .•• ; =  cos.  x.-.dsm.  x  =  cos.xdx. 

1  Ai  dx 

2.  To  differentiate  cos.  x. 

d  cos.  X  =  d  sin.  (i  *  —  x)  =  —  cos.  (|  *  —  x)  dx  =  —  sin.  x  dx. 

*  The  differentiation  of  circular  functions  may  be  obtained  independently  of 
these  results.    See  ths  note  (A)  at  the  end  of  the  volume. 


20  THB  DIFFEKENTIAL  CALCULUS. 

Co7'.  As  d  COS.  =  —  d  ver.  sin.  .•.  d  ver.  sin.  x  =  sin.  xdx. 
3.  To  differentiate  tan.  x. 

sin.  .r       COS.  r  ('  :  »n.  '■■  —  sin.  x  d  cos.  a; 


d  tan.  a?  =  d 


COS.  a;  COS.  'X 

that  is 

cos.  ^x  -\-  sin.  -X  ,  1        ,  „     , 

d  tan.  X  = :; dx  = —  dx  =  sec.  ^x  dx. 

COS.  "X  cos.  X 

4.  To  differentiate  cot.  x. 

dcot.  X  =  dtan.  (|  *  —  a;)  =  —  sec.  ^(i  ii'  —  x)  da;  =  — cosec  ^xdx. 

5.  To  differentiate  sec.  x. 

,  7      1  sin.  x    ,  , 

a  sec.  x  =  d 5-  = —  dx  =  tan.  x  sec.  x  dx. 

cos.  ■'a;       COS.  ^x 

6.  To  differentiate  cosec.  x. 

-       1  COS.  x   ,  .  , 

a  cosec.  x  =  d  — = : — —  dx  =  —  cot.  x  cosec.  x  dx. 

sm.  X  sin.  X 

Tliese  six  forma  the  student  should  endeavour  to  preserve  in  his  me- 
mory. 

EXAMPLES. 

1.  y  =^  sin.  ^a:  .*.  dy  =  2  sin.  x  d  sin.  x  =  2  sin.  x  cos.  a;  dx 
=3  sin.  2a;  dx, 

.'.  ■:ir  =  sin.  2x. 
ax 

2.  t/  =  sin. "x.'.  dy  =  nsin.  ""'  xdsin.  x  =  nsin. ""^xcos.  xdx, 

•  '  -J-  =  n  sm.      X  cos.  x. 
dx 

3.  1/  =  COS.  mx  .'.  dy  =  —  sin.  mxdmx  =  —  m  sin.  mxdx, 

dy 

.•.-r  =  —  m  sin.  mx. 
dx  ' 

4.  M  =  2/  tan.  of,  y  being  a  function  of  x, .'.  du  =  tan.  x"  dy  + 
y  d  tan.  x",  now  d  tan.  x"  —  sec.  "x"  dx"  =  nx"~'  sec.  ^x"  dx, 

dw  diy 

.*.  -7-  =  tan.  x"  T~  +  wx"~^  sec.^  x". 
ax  ax       •' 

5.  «  =  cot.  x"  .*.  du  '=  —  cosec.^  x^  dx*'.  Put  z  =  x*  .*.  log. 
2  =  1/  log.  X, 


THE  DIFFERENTIAL  CALCULUS.  21 

dz  dx       ,,77/    ^-"^ 

.:  —  =  y —  +  log.  xdij  .•.  as  =  dx^  =  {]}  —  +  'Og-  ^  dy)x^, 

Z  X  ^ 

du  „       ,1/    ,    ,  dii. 

and  —  =  —  cosec.2  a;^  (^  +  log.  x  -/)  o^. 

rfx  X  dx' 

6.  y  =  xe  "'  *  .'.  dy  =  e  "'■  "^  da;  +  xe  "'•  "^  ci  cos.  x  =  e  '''"■  * 
(1  —  X  sin.  x)  dx, 

.'.  -~  =  e  '='""  '  (1  —  sin.  a;), 
aa? 

J  (a;  e  "^  ')         ,  ,  , 

7.  y  =  log.  (x  e-^  ')  .•.dy=       ^^,,3..    >  and  d  (xe-  -)  = 

gcoB.  r  ^j  —  ^  gjj^^  a:)(Za:, 

dy  _  I  —  X  sin.  x 

'  '  dx  X 

dy  

8.  y  =  COS.  X  +  sin.  x  V  —  1,  .*.  '7Z,— — sin.a;+cos.x\/ — 1 

dy 

9.  w  =  cos.  X  -\-  COS.  2x+  COS.  3x  +  &c.  .•.  -j^  =  —  (sin.  x 

dx 

+  2  sin.  2a;  +  3  sin.  3a;  +  &c.) 

10.  y  =  xe  '^"^  '  .-.-p  =  \1  -\-  X  sec.^  xl  e  '="'•  ". 

sin. ""a;      dy  .  sin."""' a;,     ,       ,sin.'"+' a;  , 

11.  w  = —-.'.-f-  =  m\ ~-—l  +  nl -rr-i. 

cos.  a;      dx  cos.    '  a;'  'cos."'*'' a; ' 

du  lV   ,  ,  dy^ 

12-  u  =  sec.  3fi  .'.-r-  =  tan.  3^  sec.  x^  a;^J-+log.a;  —-J. 
rfa;  'a;         °      dx^ 

(15.)  In  the  preceding  trigonometrical  expressions,  the  arc  is  con- 
sidered as  the  independent  variable,  and  the  lines  sine,  cosine,  &c. 
as  functions  of  it ;  we  shall  now  consider  the  inverse  functions  as 
they  are  called,  that  is,  those  in  which  the  arc  is  considered  as  a  func- 
tion of  the  sine,  the  cosine,  &c.  A  particular  notation  has  been  pro- 
posed for  inverse  functions  :  thus,  if  j/  =  Fx  be  the  direct  function, 
then  X  =  r~^  y  is  the  inverse  function,  that  is,  if  we  represent  the 
function  that  yisofxhyy  =  Fa-,  the  function  that  x  is  of  y  will  be 
denoted  by  x  =  F~^  y.  By  thus  representing  these  inverse  functions, 
we  may  return  immediately  to  the  direct  functions,  considering,  for  the 
moment,  F~'  in  the  light  of  a  negative  power  of  F,  or  an  equivalent  to 

-^ ;  for  then  x  =  F~'  y  immediately  leads  toy  =  Fx.*     Thus,  if 
*  See  note  (B'). 


22  THE  DIFFERENTIAI.  CALCULUS. 

X  =  log.z' y  '"'  y  =  log.  X,  the  inverse  function  log."'  y  meaning 
the  number  whose  log.  is  y.  In  like  manner,  y  =  sin.~'  x  means  that 
y  is  the  arc  whose  sine  is  x ;  that  is,  returning  to  the  direct  function, 
sin.  y  =  X. 

1.  To  differentiate  y  =  sin.""'  x. 

Here  the  direct  function  is  sin.  y  =  x  .♦.  d  sin.  y  =  dx,  that  is, 

J         J         dy  I  1  1 

cos.  t/rfw  =  dx  .:  -r-  = =  — —  =  —  . 

dx       cos.y       VI  — sin.  =4/       ^  1  —  x" 

2.  To  differentiate  y  =  cos.~'  x. 
cos.y  =  X  .•.  sin.  ydy  =  dx  .•.  Ji  =.  — 


dx  sm.  y  y/l—cos.^y 

3.  To  differentiate  y  ==■  versin."'  x. 

versin.  y  =  x  .'.  sin.  ydy  =  dx, 

"  dx        sin.y         y/2x  —  x' 

4.  To  differentiate  y  =  tan."'  x. 

tan.  y  =■  X  .•.  sec.^  Ww  =  dx  .*.  -^^  = r—  =  — ; -. 

^  ^  ^  dx       sec^y       1  +  x' 

5.  To  differentiate  y  =  cot."'  x. 
)t.  y  =  a:.'.  —  cosec'^ydy  =  dx.*. 

6.  To  differentiate  y  =  sec."'  x. 


,    1         ,        dy  1  —1 

cot.  V  =  x.-.  —  cosec.ydy  =  dx.'.-r-  — s"  = ; i* 

^  ^  ^  £f.i!  cosec.  ""y       1  +  x" 


sec.  y  =  X, .'.  tan.  y  sec.  yc?i/ 

1 


=  dx...^  = 


1 


dx      tan.  t/  sec.  y 


X  -v/x*  —  1 
7.  To  differentiate  ?/  =  cosec."'  x. 

cosec.  t/  =  X  .*.  —  cotan.  1/sec.  i/(ij/  =  dx  .'.-j-  — 


dx      cot.  J/  sec.  y 


\.  y  =■  sin. 


X  -v/x^ — 1 

EXAMPLES. 

%      _                 1 

dmx        VI  _  ,^3;c3 

*  dx 

VI  —  mV 


THE    DIFFERENTIAL    CALCULUS.  23 

fit  I  fi  siTi  —  V^ 

2.  y  -  X  sin.~'  sp  .'.  -~  =  sin.  '  ar^  +  x ,- and 

ax  ax 

d  sin.  '  a^  dx 


dx  VI—  X* 

dy         .      ,    o  ,         2x2 


3.  y  =  COS."'  X  \/ 1  —  x".     Put  X  Vl  —  x^  =  z 

— dz 
.'.  dy  =  — — 

VI    —    2^ 
X^ 


hutdz-{Vl—x'  —  —==)dx,iind  Vl  —  z^  =  Vl  — x^  +  x' 
V  1  —  ar' 

^    dy  _  —  1  +  2x^ 

*  *  dx        ^(1  _a;a_|-a;*)  (1  —^' 

4.  y  =  tan.~'  -. 
^  2 


rfy  _  1  .   dy  _ 


8 


i^x       ^  a^     '  '  dx        4  +  x^  * 

6.  J/  =  cot.-'(a  +  mx)='  .:dy  =  —  ^_^^^_^^^^y  rf (a  +  wx)^. 

J  /      I         va       n  /      t        \     J        ^y       2m  (a  +  mx) 

d  (a  +  mxY  =  2  (a  +  wx)  mdx  .-.  -r-  =  — ~ — ; 7. 

dx       14-(a  +  nix)* 

6.  y  =  sec."'  —  .:  dy= —  d  — ,  and 

^  X"         ^  la  x" 

<.^(^)"  - 1 

-  a  am  dx 


x"  x'^' 


*  *  dx  ^^ 


_,  VI  +  x^        ,                VI  +  r'    . 
7.  »/  =  cosec.  ' . •.  dy  =  —  d — 


VI  +  x'     1  +  r"  1 


24  THE  DIFFERENTIAL  CALCULUS, 


—  d ,  but  a 

X  ar  X 

1 


ar  v/ 1  +  or 


dx, 


dy  _        1 

'  '  dx        1  +   x^' 

8.  y  =  (sin.-^  xY  .'.-i^  =  2  sin.-'  a?  — r,. 

,  ^  */  1 

9.  y  =  cos.~  •         ~ 


Vl   +  r'       dx       1  +  X- 


,        _  .      _i     ,'  •  —  X       dy  \ 

1.  y  =  tan.      v/— -—  -'' -r  =  —  ^  ,T- 


1    +   x       dx  2n/1  —  x^ 

n.V  =  (cot.-'.)^-.|  =  -^oot.-.. 

12.  J/  =  sec.-'x".'.        — 


dx  X\/ 3?"  —  1 


,_  .      ^       d\\ 

13.  «  =  cosec.     mxr  .',  -t-  =■ 


dx       xVm^x^  —  1 


•       1  dii  ^ 

"^  dx       \/2  —  e^ 

(16.)  In  the  preceding  expressions  the  radius  of  the  arc  is  always 
represented  by  unity,  but,  as  the  differentials  are  frequently  required 
to  radius  r,  we  shall  terminate  this  chapter  with  the  several  formulas 
in  (15)  accommodated  to  this  radius.     We  must  observe,  that  as  y 

y 

and  X  are  homogeneous  in  each  of  those  forms,  -  is  always  a  num- 

dv 
ber,  so  that  this  ratio  in  the  limit,  that  is  -7^,  is  a  number.     Hence,  r 

dx 

must  be  introduced  as  a  multiplier  so  as  to  render  the  numerator  and 
denominator  of  each  expression  of  the  same  dimensions.  The  for- 
mulas, therefore,  become 

rdx 

d  sin.~'  X  =     ■  „  =• 


THE  DIFFERENTIAL  CALCULUS.  25 

rdx 


d  cos.~*  X 


Vi'^  —  ar' 

rdx 
d  versin.~'  x  =  =• 

V  2rx  —  x^ 

r'dx 
r^dx 


d  tan.~^  X  = 
d  cot.~*  X  = 


T^  +  a^ 

r'dx 
d  see."'  s 


x\/  x^  —  r^ 

r^dx 
d  cosec."'  X  =  — 


xy/ar^  —  r^ 

On  successive  Differentiation. 

(17.)  Since  the  differential  coefficient  derived  from  any  function 
of  a  variable  may*  contain  that  variable,  this  coefficient  itself  may  be 
differentiated,  and  we  thus  derive  a  second  differential  coefficient.  In 
like  manner,  by  differentiating  this  second  coefficient,  if  the  variable 
still  enters  it,  we  obtain  a  third  differeiitial  coefficient,  and  in  this  way 
we  may  continue  the  successive  differentiation  till  we  arrive  at  a  co- 
efficient without  the  variable,  when  the  process  must  terminate. 

Thus,  taking  the  function  y  =  ax'*,  we  have,  for  the  first  differ- 

dy 
ential  coefficient,  -p  =  4ax',  as  this  coefficient  contains  x,  we  have, 

by  differentiating  it,  the  second  differential  coefficient  =  12ax^  ; 
continuing  the  process,  we  have  24ax  for  the  third  differential  coeffi- 
cient, and  24a  for  the  fourth,  which  being  constant  its  differential 
coefficient  is  0. 

If  we  were  to  express  these  several  coefficients  agreeably  to  the 
notation  hitherto  adopted,  they  would  be 

first  diff.  coef.  -~  =  4cw^ 
dx 

d^ 
dx 
second  diff.  coeC — -^ =  12ax^, 

*  It  must  contain  the  variable,  unless  in  the  single  case  of  its  being  constant. 

Ed. 

4 


26  THE  DIPFEllENTIAL  CALCULUS. 


d 


d^ 

dx 
dx 


third  diff.  coef. -i =  24  ax, 

&c. 
But  this  mode  of  expressing  the  successive  coefficients  is  obviously 
very  inconvenient,  and  they  are  accordingly  written  in  the  following 
naore  commodious  manner : 


first  difi*.  coef. 

second  diff.  coef. 

third  diff.  coef. 

nth  diff.  coef. 


dx 

dx'' 

dx"* 
d"y 


in  which  notation  it  is  to  be  observed,  that  d^,  d^,  &c.  are  not  powers 
but  symbols,  standing  in  place  of  the  words  second  differential,  third 
differential,  &c.  The  expressgions  dx'^,  daP,  &c.  are  on  the  contrary 
powers,  not,  however,  of  x,  but  of  rfx  :  to  distinguish  the  differential 
of  a  power  from  the  power  of  differential,  a  dct  is  placed  in  the  former 
case  between  d  and  the  power. 

(18.)  The  following  are  a  few  illustrations  of  the  process  of  suc- 
cessive differentiation : 


I.  y  =  if". 


dx 

d3» 

dPy 

~d^ 

d'y 


=  mx*-', 

=  OT  (m —  1)  x'^S  ^ 

=  m  [m  —  1)  (tn  —  2)  x~~', 

=  m  (m  —  1)  (wi  —  2)  (m  —  3)  x**-% 


rfx« 

&c.  &c. 

2.  tt  =  yz,  both  y  and  z  being  functions  of  x, 


THE  DIFFERENTIAL  CALCULUS. 


27 


du 

dx 

dr-u 
1^ 


d\ 


dz 


dy 


dx 

dr-z 


dx" 
d^z 


=  y 


dx" 

d'^z 


dx" 

'  dx^ 
&c. 

3. 

y 

=  log. 

X 

dy 
dx 

= 

1 

x 

d^y 
dx" 

= 

1 

d?y 
dx" 

= 

2 

4. 

y 

=  (f. 

dy 
dx 

= 

e' 

d-y 

= 

e' 

dx 
dydz 
dx" 
dydz 
~d^ 
dyd^z 


zi- 


y 


dx" 


+  3 


dx" 

dr" 
dzdPy 
~d^ 

&c. 


,dzdy_ 
dx^ 


+ 


dx" 


&c. 


d'^y  _       2-3 

dx"^  X* 

dhj  _  2  .  3  •  4 

doc"  XT' 

dHj  _  _  2  •  3  •  4 

5^  ^ 


dx" 

d'y  -  ^ 


dx' 


&c. 


If  instead  of  c  the  base  were  a,  the  several  coefficients  would  be 
log.  a  '  a",  log.^a  •  a%  log.^a .  a%  log.^a  •  a',  &c. 

It  appears,  therefore,  that  exponental  functions  possess  this  property, 

d"y 
VIZ.  that  -r^  -^  y  ia  always  constant. 

5.  y  =  sin.  x. 

'^y  _ 


dx 
di? 


cos.  X 


=  —  sm.  X 


dx' 
dx* 


cos.  X 


sm.  X 


&c. 


We  need  not  multiply  examples  here,  as  the  process  of  successive 
differentiation  will  be  very  frequently  employed  in  the  next  two 
chapters. 


28  THE  DIFFERENTIAL  CALCULUS. 


CBAPTSIS  IIX. 

ON  MACLAURIN'S  THEOREM. 

(19.)  Ify  represent  a  function  of  x,  which  it  is  possible  to  develop 
in  a  series  of  positive  ascending  powers  of  that  variable,  then  will 
that  development  be 

where  the  brackets  are  intended  to  intimate  that  the  functions  which 
they  enclose  are  to  be  taken  in  that  particular  state,  arising  from 
taking  x  =  0.* 

For,  since  by  hypothesis 

y  =  A  +  Bx-{-    Car'+         Da;^  +  Ex*  +  &c...(l> 

...  J^=  B    +  2Cx  +       3Dx2+  4-EaP+Sic. 

ax 

■j^  =  2C     +  2  •  ZBx  +      3  .  4Er»  +  &c. 

erar 


3D     +2.3. 4Ex  +  &c. 


^  _ 

dsP   ~ 

&c.  &c. 

Let,  now,  a:  =  0,  then 

dy 

rg-]=2.3D...D=-i-rg-] 

Ldx^-"  2-3  LrfariJ 

&c.  &c. 

*  This  plan  of  enclosing  the  differential  coefficient  in  brackets  we  shall  usually 
adopt,  when  wc  wish  to  express  not  the  general  state  of  this  function,  but  that 
state  which  arises  from  the  variable  taking  a  particular  value.  What  that  value 
is  will  generally  be  made  known  by  the  nature  of  the  inquiry. 


THE    DIFFERENTIAL    CALCULUS.  29 

Hence,  by  substitution,  equation  (1)  becomes 

»  =  W  +  [|]^+i[^F+^3[g-]-'+&c (2), 

which  is  Maclaurin's  theorem  for  the  development  of  a  function, 
according  to  the  ascending  powers  of  the  variable.  We  shall  apply 
it  to  some  examples. 

EXAMPLES. 

(20.)  1.  Let  it  be  required  to  develop  (a  +  x)",  the  exponent  n 
being  any  uumber  whatever,  either  positive  or  negative,  whole  or 
fractional,  rational  or  irrational. 

Put  y  =  {a  +  x)"     .     .     therefore     .      [j/]     =  a" 

...  A  =  n(a  +  x)"-' r-^]  =  na'^' 

dx  ^  dx 

^=.n{n-\){a+xr-'      .     .     .  [^-]  =  n(n- !)«-' 

g-  =  n(n-l)(n-2)(a  +  x)-     [g.]  = 

n(n— 1)  {n  —  2)dr^ 
&c.  &c. 

Substituting  these  values  for  the  coefficients  in  the  foregoing  theo- 
rem, there  results 

(a  +  xY  =  a"  +  »a"-'  x  +  — ^ a      x^  +  — ^^ — '- 

2i  2 '  3 

a"-3  x^"  +  &c. 

and  thus  the  truth  of  the  Binomial  Theorem  is  established  in  its 

utmost  generahty. 

2.  To  develop  log.  {a  +  x). 

Put  y  =  log.  (o  -|-  x),  therefore  [t/]   =  log.  a 

d^i/     _       1 

•  ^dar"  ^  ^ 

rf'y     _      2.3 

•  trf^]  "^ 

&c. 


dx 

^y  _ 

1 

^y  _ 

(a  +  xf 
2 

(a  +  xr 

2-3 

dx' 
&c. 

(a  +  X)* 

30  THE  DIFFERENTIAL  CALCULUS. 

...  lo..  ia  +  .)  =  log.  a+^_^+^_^+&c. 

dv 
If  7/  =  log.  X  were  proposed,  then,  since  [y],  [^r;]*  &c.  are  infinite, 

we  infer,  for  reasons  similar  to  those  assigned  at  art.  4,  that  the  de- 
velopment in  the  proposed  form  is  impossible. 
3.  To  develop  sin.  x. 

y    =  sin.  X        ....       [?/]    =  0 

i  =  COS.  X         ....[-^1=1 
dx  dx 

dhi  .  dii^  ^ 

^  =  -co..x    .    .    .    .[^]=-l 

^  =  sin..         .     .     .     .  [^J=0 

dx*  Hx*  -■ 


dhi  ^dhj 

&c.  &c. 


.  •.  sin.  z  =  X -\ &c. 

1-2-31-2.3-4-5 

4.  To  develop  cos.  x. 

y    =  cos.  X         ....       [?/]   =  1 

%=_.„..     .     .     .     .[^]  =  o 

^=—    •    •    •[^]=-' 

&c.  &c. 

x^  x^ 

•  •.  cos.  X  =  1 ^ + — - — : &c. 

1-2         1-2-3-4 

6.  To  develop  a'. 

y    =  a''        therefore  [y]    =  1 

^  =  *--  •  •  •[f]  =  - 

*  A  is  put  here  for  the  hyperboHc  logarithm  of  the  base  a,  that  is,  for  the  ex- 
pression 

(a  —  1)  —  i  (a  —  1)2  +  J  («  —  1)='  —  ^- 


THE  DIFFERENTIAL  CALCULUS.  31 

^-Aa      ....     L^^J 

— -Aa      .     .     .     .     Lrf^J       A 

&c.  &c. 

,    .      ,    AV    ,      AV      ,    ^ 
.•.a^  =  l+A.  +  ^^  +  P^3  +  &c. 

which  is  the  Exponential  Theorem. 

Since  A  =  log.  a,  we  may  give  to  the  development  the  form 

a'  =  1  +  X  log.  a  +  -{x  log.  af  +  r-^  (x  log.  a)=^  +  &c. 

For  a:  =  1,  we  have  the  following  expression  for  any  number,  a,  in 
terms  of  its  Napierian  logarithm  : 

1  1 

a=  1  +  log.  a  + 2^°S-^"  +  2 — 3^^^'^"'  "^  ^^' 

changing  a  into  the  Napierian  base,  e,  we  have 

a^  aP 

e-  =  i+a;+—   +   ^— ^  +  &c. 

which,  when  x  =  1,  gives,  for  the  base  e,  the  value 

c  =  1  +  1  +  2  +  2T3  +  ^*^- 

(21.)  From  the  development  of  e'^  may  be  immediately  derived 
several  very  curious  and  useful  analytical  formulas,  and  we  shall  avail 
ourselves  of  this  opportunity  to  present  the  principal  ones  to  the  no- 
tice of  the  student. 

If,  in  the  development  of  e',  we  put  zV  —  1  for  x,  we  shall  have 

_  z^         zW  —  I  z" 

«'^-'  =  1  +  ^^-1 -r^-1  .  2.  3 +1.2. 3-4  +  ^^- 

and,  changing  the  sign  of  the  radical, 


z"         z"  V  —  I 


«  i       z  y/        1^       1  •  2^1  •  2  •  3^1-2-3-4     *^^* 

If  these  expressions  be  first  added  and  then  subtracted,  there  will  re- 
sult the  following  remarkable  developments,  viz. 


32  THE  DlFFliHENTIAL  CALCULUS. 


W-i  4.   gW-i  ^ 


=  1  —  1 ^  +  ^i — 7. ^— T  —  &C. 


2  l-2'l-2-3-4 


+  ,      ^     o     . — £  —  &C. 


2^'^[  l-2-3'l-2-3-4-5 

Now  it  has  been  seen  (examples  4  and  3)  that  these  two  series  are 
also  the  respective  developments  of  cos.  z  and  sin.  z ;  hence,  putting 
X  instead  of «,  we  may  conclude  that 

gS-V  — 1   g-»V  — 1 

sin.  X  = := ....  (1) 

2^/  —  1 

cos.  X  = z ....  (2) 

where  the  sine  and  cosine  of  a  real  arc  are  expressed  by  imaginary 
exponentials. 

These  expressions  were  first  deduced  by  Euler,  and  are  consider- 
ed by  Lagrange  as  among  the  finest  analytical  discoveries  of  the  age. 
(Calcul  des  Fonctions,  page  114.) 

(22.)  If  for  the  real  arc  x  we  substitute  the  imaginary  arc  x\/  —  1 

we  shall  have 

e~^  —  e* 


sin.  (a:v/-l)   =  ^-^==....(3) 

c-^  +  e^ 

COS.  {x  V  —  1)   =  ^ ....  (4) 

Sin. 
Also,  since  — '-  =  tan.,  it  follows,  from  (1)  and  (2),  that 

COS. 

C^s/  —  ^  g— jV  — 1         g^i^^Ti  J 

y/  —  1  tan.  X  =  — = =  = =r * 

gTV-i  -f  e-^^/-'       e^-^-^  +  1 

By  multiplying  equation  (1)  by  ±  V — 1,  and  adding  the  result  to  (2), 
we  have 

COS.  X  ±  sin.  X  V  —  1  =  c±'^-'  ....  (6;) 
or  if  we  change  x  into  mx, 

cos.  mx  ±  sin.  mx  \/  —  1  =  c±"«^'-'  ....  (6), 
but  e±""^~'  is  c^''-^'"'  raised  to  the  mth  power.     Hence  this  singu- 
lar property,  viz. 

*  Multiplying  the  numerator  and  denominator  of  the  second  member  of  the 
equation  by  e*^—'.  Ed. 


THE  DIFFERENTIAL  CALCULUS.  89' 


(cos.  X  ±  sin.  X  ■«/  —  1)"'  =  COS.  mx  ±  sin.  mx  V  —  1  ....  (7), 

which  was  discovered  by  de  Moivre,  and  is  hence  called  De  JVLoivre'a 
formula. 

If  the  first  side  of  this  equation  be  developed  by  the  binomial  theo- 
rem, it  becomes 

,   m(m  —  1)  ,     „        „ 

cos,  '"x  ±  m  cos.  "*-*  xp  -\ ^^ cos.  ""-^  xp^  it  &c. 


p  being  put  for  the  imaginary  v/  —  1  sin.  .r. 

Now  in  any  equation,  the  imaginaries  on  one  side  are  equal  to  those 

on  the  other,  {Algebra)  ;  hence,  expunging  from  this  expression 

all  the  imaginaries,  that  is,  all  the  terms  containing  the  odd  powers  of 

p,  we  have,  in  virtue  of  (7), 

m  (in  —  1) 
cos.  mx  =  COS.  "x  — r COS.  "'"^  X  sin.^ar  + 

m{m  —  1)  (m  —  2)  {in  —  3 

n — o — A cos.  "^^  X  sin.  *x  —  &c. 

2  •  3  •  4 


In  like  manner,  equating  sin.  mx  y/  —  1  with  the  imaginary  part  of 

the  above  development,  and  then  dividing  by  ■«/  —  1 ,  we  have 

m{m — l)(m — 2) 
sm.  mx=m  cos.*"" '  a:  sin.  x  — oT^ cos."""  ^xsm.^x-\-  &c. 

From  these  two,  series  the  sine  and  cosine  of  a  multiple  arc  may  be  de- 
termined from  the  sine  and  cosine  of  the  arc  itself 

(2ft)  If  in  the  formula  (2)  we  represent  e''^~^  by  y,  then  c~'^~' 
=  - ;  therefore, 

y 

1 

2  cos.  a:  =  V  +  ~ 

a  y 

or,  if  in  the  same  formula  mx  be  put  for  x,  we  have 

1 
2  COS.  mx  =  V"  H — — 

3  ym 

and  from  these  two  equations  we  deduce  the  following,  viz. 
y^  —  2y  COS.  a:  +  1  =  0  ....   (1) 
«/**  —  2r/'"  COS.  mx  +  1  =  0  .   .  .  .   (2). 
Since  these  equations  exist  simultaneously,  the  latter  must  have 
two  of  its  roots  or  values  of  y  equal  to  the  two  roofs  of  the  former, 


34 


THE  DIFFERENTIAL  CALCULUS. 


and  must,  therefore,  be  divisible  by  it ;  or,  putting  6  for  mx,  we  have 

y^n  _  2y^ioS.  d  +   1    =  0    .    .    .     .     (3), 

divisible  by 


0 


(4). 


ys  —  2y  cos.  —  +  1 

But  cos.  6  =  COS.  {&  +  Snir),  n  being  any  whole  number,  and  if  = 
180° ;  hence,  making  successively  »  =  0,  =  1,  =  2,  &c,  to  n  = 
m  —  1,  we  have,  since  the  first  equation  continues  to  be  divisible  by 
the  second  in  these  cases, 


y2m  —  2?/'"  cos.^  +  1  =  (j/^  —  2j/  cos. 


+  1) 


6  +  2* 
X(y'  — 2J/C0S.— ^^+  I) 

6  +  4* 
X  {y'—2ticos.—^^+  1) 

^  +  6* 
X  (tf — 2ycos. \r  1)  &c.  to  m  factors. 

The  truth  of  this  equation  is  obvious,  for,  while  the  substitution  of 
6  +  2*111'  for  6  causes  no  alteration  in  the  expression  (3),  the  same 
substitution  in  (4)  gives  to  that  expression  a  new  value,  for  every  va- 

lue  of  n,  from  7»  =  0  to  n  =  m  —  1,  for  the  arcs  — , &c.  are 

m        m 

all  different.     As,  therefore,  the  expression  (3)  is  divisible  by  (4) 

under  all  these  m  cheuiges  of  value,  it  is  plain  that  these  are  its  in 

quadratic  factors. 

In  this  way  may  any  trinomial  of  the  form  i/^  —  2ky'"  +  1  be  de- 
composed into  its  quadratic  factors,  provided  k  does  not  exceed  unity, 
for  then  k  may  always  be  replaced  by  the  cosine  of  an  arc. 

(24.)  The  geometrical  interpretation  of  the  foregoing  equation, 
presents  a  curious  property  of  the  circle,  first  discovered  by  De 


■B  'B 


Moivre.  To  exhibit  this  property,  let  P  be  any  point 
either  within  or  without  the  circle  whose  centre  is  0,  and 
let  the  circumference  be  divided  into  any  number  of 
equal  parts,  commencing  at  any  point  A,  Join  the  points 
of  division,  A,  B,  C,  &c.  to  P,  then,  since  in  the  fore- 


THE  DIFFERENTIAL  CALCULUS.  35 

going  analytical  expression  the  radius  OA  is  ex- 
pressed by  unity,  we  shall  have,  by  introducing 
the  radius  itself  so  as  to  render  the  terms  homo- 
geneous, the  following  geometrical  values  of  the 
above  factors,  where  it  is  to  be  observed  that 

Z  POA  =  —  and  OP  =  y, 
m 

yim  _  2ym  cos.  6  -f  1  =  OF^  —  20?""  X  OA"  COS.  m  ( AOP)  +  AO^" 

t/«  —  2y  COS. f-  1  =  0P2  —  20P  X  OA  cos.  AOP  +  AO^  =  PA»* 

111 

y^  —  2y  cos.  ?-±-!l  -|-  i  =  0P«  —  20P  X  OA  cos.  BOP  -|-  B0«  =  PB« 
m 

y«  —  2«  cos.  ^-iil  +  1  =  0P«  —  20P  X  OA  cos.  COP  +  C0»  =  PC« 

in 

&c.  &c  &c 

Hence, 

OP*"*  —  SOP"*  X  0A"»  cos.  m  (AOP)  -\-  OA^-"  =  PA«  X  PB«  X  PC*  X  &.c. 
and  this  is  Demoivre's  property  of  the  circle. 

(25.)  If  AOP  =  0,  that  is,  if  P  be  upon  the  radius  through  one  of 
the  points  of  division  A,  then  cos.  m  (AOP)  =  1.     Hence, 

OP**  —  20P'"  X  OA""  +  OA'"*  =  PA^  X  PB"  x  PC^  x  &c. 
consequently,  extracting  the  square  root  of  each  member, 

OP"*  ^  OA'"  =  PA  x  PB  X  PC  x  &c. 
If  the  arcs  AB,  BC,  &c.  be  bisected  by  A',  B',  &c.  the  circumfer- 
ence will  be  divided  into  2m  equal  parts,  and,  by  the  equation  just 
deduced 

0P=^  ^  OA^™  =  PA  X  PA'  X  PB  X  PB'  x  &c. 
that  is 

0?=^"*  ^  0A=^  =  (OP"*  ^  OA-")  PA'  X  PB'  X  &c. 
therefore, 

Qpan         OA'^ 

__L — __-  =  OP"*  +  OA"*  =  PA'  X  PB'  X  PC  X  &c. 

and  these  are  Coles's  properties  of  the  circle. 

(26.)  If  now  we  return  to  the  expression  (6),  and  suppose  x  = 

— ,  it  becomes 

♦  Gregory's  Trigonometry,  p.  54,  or  LacroLx'a  Trigonometry. 


»   V) 


36  THE  DIFFERENTIAL  CALCULUS. 


7  _1  =ca>/-i...log,  n/  — 1    =   ~^-^\, 
and 

From  the  second  of  these  equations  we  get 


It 


=  21ogV  — 1  _  log.(v/  — 1)^   ^  log.— 1  ^  _  ^^TT 


^/  —  l  V  — 1  V  — 1 

log.  —  1. 
From  the  third 

,    , , —  *  1  flr^  I  <it^ 

(v'-l)-'-.  =  i_-  +  3-^.-_^^^.-  +  &c. 

Two  very  singular  results ;  first  obtained  by  John  Bernouilli. 
6.  To  develop  tan.  x. 

y    =  tan.  x,  therefore     ...       [y]     =  0 

-/-  =  sec.2  X ri  =  1 

ax  ^  dx-^ 

^  =  2sec.».ta„.x     .     .     .     [§]  =  0 

g-  =  2sec.=  x(l  +  3lan.".).     [§]  =  2 
We  thus  see  that  the  first  two  terms  of  the  development  are  x  + 
,  but  we  shall  not  continue  the  differentiation,  since  it  does 


1  •2-3 

not  make  known  the  law  of  the  series.  The  development  will  be 
more  readily  obtained  by  means  of  those  already  given  for  sin.  x  and 
COS.  X,  as  follows : 

x"*  X* 

^~1  •2-3   "^    1  •2-3-4-6~^*^' 
tan.  X  = T 

X-  x"* 

1--^—    +      1  •  2  .  3  •  4   -  ^^- 
therefore  the  development,  found  by  actual  division,  is 

2x'^        ,  16a^'  „ 

tan.  X  =  X + isc. 

1-2-3  ^  1-2  -3 -4  -5 

but,  to  obtain  the  law  of  the  terms  and  thus  be  enabled  to  continue 

them  at  pleasure,  it  will  be  best  to  apply  the  method  of  indeterminate 

coefficients.     Assume,  therefore,  this  fraction  equal  to  the  series 

A,  .r  +  A3  r'  +  As  a^  +  A7  x^  +  &c. 


THE  DIFFERENTIAL  CALCULUS. 


37 


Multiplying  this  by  the  denominator,  we  have  this  expression  for  the 
numerator,  viz. 

a?  3^ 


+ 


l-2'3        1.2-3-4-5 


Ai  X  +  A3 
A, 


1  -2 


3P  + 


+ 


1  •  2 

A, 


&c  = 


x5+  &c. 


1  -2  -3  •  4 

Hence,  equating  the  coefficients  of  the  like  terms,  there  results 
A,  =  1  therefore  Ai  =  1 


A, 


A, 


1  •  2 


1  -2  -3 


A3    = 


1-2        1-2-3 


A. 


+ 


1-2-3 
1 


1-2        1-2  -3  -4       1-2-3-4-5 
1 


,A5  = 


A. 


1-2       1  •2-3'4 


+ 


1-2 • 3-4-5 
&c.  &c. 

the  law  of  these  coefficients  being  such,  that 

A2„_i  A2„_3  ,  ,  Ai 


A2_4.|    


1-2        1  •2-3-4 


.   .  .   .    ± 


1-2.3 


2» 


1  -2  -3  .  .  (2rt  +  1) 
7.  To  develop  tan.~'  x. 
y    =  tan.~^  x     .     .     .     .  therefore  .     . 

dy  _      1 

dx        1   +  x^ 

cPy   _  2x 

dJ  "^  ~ 

2    '4:3? 


+ 


(1  +  x^) 
d^y   _  2x 

dx^            (TT^'  '    {I  +  ^) 

d'y ^x  2'x 2^-3ar^ 

d^  ~  (1  +  x'f    (1  +  0^)3""  (r+  ^) 


dy 
dx 


[^]  =  > 


Mr' J 


38  THE    DIFFERENTIAL    CALCULUS. 

dx"        (1  +  a:^)^       (1  +  x^Y       (1  +  x")' '     '  Mx*  ^ 
&c.  &c. 

.'.  y  =  tan.  y  —  i  tan.  ^y  +  |  tan.  ^y  —  \  tan.  'y  +  &c. 
If  1/  =  45°,  then  tan.  y  —  1 ; 

.-.  arc  45°  =  1  —  i  +  1  — -|  +  &c. 
(27.)  From  this  series  an  approximation  may  be  made  to  the  cir- 
cumference of  a  circle,  but,  from  its  very  slow  convergency,  it  is  not 
eligible  for  this  purpose.  Euler  has  obtained  from  the  above  general 
development  a  series  much  more  suitable,  by  help  of  the  known  for- 
mula, {Gregory's  Trig.,  page  46,) 

,    , ,       tan.  a  -f  tan.  6* 

tan.  (a  +  b)= =• 

1  —  tan.  a  tan.  b 

for,  when  a  +  6  =  45°,  tan.  (a  -f-  6)  =  1 ;  therefore, 

tan.  a  +  tan.  6  =  1  —  tan.  a  tan.  6. 

If  either  tan.  a  or  tan.  6  were  given,  the  other  would  be  determinable 

from  this  equation.     Thus,  if  we  suppose, 

^   *u      1  j_  *       I.       -i        ^^^'  ^        .       I       « —  1 
tan.  a  =  -,  then  -  +  tan.  6=1 ,  .♦.  tan.  6  = . 

n  n  n  »  +  1 

Now  the  value  of  n  is  arbitrary,  and  our  object  is  to  assume  it  so 
that  the  sum  of  the  series,  expressing  the  arcs  a,  6,  in  terms  of  their 
tangents,  may  be  the  most  convergent.  This  value  appears  to  be 
n  =  2,  or  n  =  3  ;  therefore,  taking  w  =  2,  we  have 

tan.  a  =  1,  tan.  6  =  1. 
Hence,  substituting  in  the  general  development  a  for  y  and  \  for  tan.  y, 
and  then  again  6  for  y  and  \  for  tan.  6,  the  sum  of  the  resulting  series 
will  express  the  length  of  the  arc  a  +  6  =  45°,  that  is 

arc.  45°  =  ^-3-^5  +  5^  "  7"^  +  &C- 

+  1 L_+_J ^+&c 

2       3  .  3^  ^  5  .  3*       7.3^^ 

(28.)  Another  form  of  development,  still  more  convergent  than 

thisi  has  been  obtained  by  M.  Berirand  from  the  formula 

2  tan. a 

tan.  2a  = — 

1  —  tan.  "a 

*  Lacroix  Trigonometry. 


THE    DIFFERENTIAL    CALCULUS.  89 

For  put  tan.  a  =  \,  then  tan.  2a  =  yS_,  therefore  2a  Z  45°,  because 
tan.  45°  =  1  :  from  this  value  of  2a  we  deduce 

2  tan.  2a  120 

tan.  4a  = —  = 

1— tan.='2a       119 

.-.  4a  7  45°. 

Let  now  4a  =  A,  45°  =B,  A  —  B  =  b=  excess  of  4a  above  45°, 

then  we  have  45°  =  A  —  6.     But 

.        UN       ^       ,         tan.  A  —  tan.  B  1 

tan.  (A  —  B)  =  tan.  6  =  — ; =r  =  — - 

^  ^  1  +  tan.  A  tan.  B       239 

Consequently,  if  in  the  general  development  we  replace  yhy  a  and 
tan.  y  by  i,  and  then  multiply  by  4,  we  shall  have  the  length  of  the 
arc  4a,  and,  since  this  arc  exceeds  45°  by  the  arc  6,  if  we  subtract 
the  development  of  this  latter,  which  is  given  by  substituting  ^i^  for 
tan.  y,  the  remainder  will  be  the  true  development  of  45°.     Thus 

45°  =4(i —  +  —- —+  kc.) 

^2       3  •  6=*       5  •  5^       7  •  5^  ^         ^ 

^(— 1 1 1 &c.) 

^239       3  •  239=*  ^  6  •  239^  ^ 

This  series  is  very  convergent,  and,  by  taking  about  8  terms  in 
the  first  row  and  3  in  the  second,  we  find,  for  the  length  of  the  semi- 
circle, the  following  value,  viz. 

If  =  3- 141592653589793. 
If  we  take  but  three  terms  of  the  first  and  only  one  of  the  second,  we 
shall  have  -r  =  3  •  1416,  the  approximation  usually  employed  in 
practice. 

(29.)  The  following  examples  are  subjoined  for  the  exercise  of 
the  student : 

8.  To  develop  y  =  sin.~^  x. 

,    sin.^  y     ,        3^  sin.'  y       ,        3^  •  5^  sin.'  y 
y  =  sm.  y-{-~—^-  +  ^    ^    .,    ,    ^  +  ^ 


1-2-3        1-2-3-4.5        1-2-3-4-5-6-7 
9.  To  develop  y  =  cos.  ~'  x. 

..  —  i  cos.^  y  3^  cos.^  y        ,    „ 

y  =  i  -r  —  COS. « ^ 2 u  &c. 

"  ^       1-2-3        l-2-3-4-5^ 

10.  To  develop  y  =  cot.  x  by  the  method  of  indeterminate  coef- 
ficients, as  in  example  6. 


40 


cot.  X  = 


THE  DIFFERENTIAL  CALCULUS. 

1        X  x^  It" 


X       3        3^-5       3^  •  5  •  7 
11.  To  develop  j/  =  (a  +  6t  +  c^r^  -f  &c.)"' 
{a-\-hx  ■\-  ex"  +  &c.)"  = 

n(n— 1)  (?i  — 2) 


&C. 


,  ,      ,    n  (n  —  ] )        ,    „ 
-\-  na"-^  c 


x^-\- 


an-3  1,3 
2-3 

4-n  (n— 1)  "-2  6c 
-|-  no"— 1  erf 


a;'-J-&c. 


This  is  the  multinomial  theorem  of  De  Moivre.     It  is  given  in  a 
very  convenient  practical  form  in  my  Treatise  on  Algebra. 


OHAFTEH  IT. 

ON   TAYLOR'S  THEOREM,  AND   ON  THE   DIFFER- 
ENTIATION AND  DEVELOPMENT  OF  IMPLICIT 
FUNCTIONS. 

(30.)  In  the  second  chapter  we  established  the  form  of  the  gene- 
ral development  of  the  function  F  (;r  +  h)'  We  here  propose  to 
investigate  Taylor's  theorem,  which  is  an  expression  exhibiting  the 
actual  development  of  the  same  function.  The  following  lemma, 
must,  however,  be  premised,  viz.  that  if  in  any  function  of  p  +  q  one 
of  the  quantities  p,  q,  is  variable,  and  the  other  constant,  we  may  de- 
termine the  several  differential  coefficients,  without  inquiring  which 
is  the  constant  and  which  the  variable,  for  these  coefficients  will  be 
the  same,  whichever  be  variable.  This  principle  is  almost  axiomatic. 
For  as  the  function  contains  but  one  variable  we  may  put  p  +  g  =  x 
or  F  (p  +  q)  =  Fx,  and  whichever  of  the  parts  p,  q,  takes  the  in- 
crement h,  the  result  ¥  (^x  -\-  h)  is  necessarily  the  same  ;  hence  the 
development  of  this  function  is  the  same  on  either  hypothesis,  and 
therefore  the  second  term  of  that  development,  and  hence  also  the 
differential  coefficient.  The  first  differential  coefficient  being  the 
same,  the  succeeding  must  be  the  same  ;  therefore  generally 
d"F  (p  +  q)        rf-F  ip  +  q) 


dp" 
whatever  be  the  value  of  n. 


dq" 


THE  DIFFERENTIAL  CALCULUS.  41 

Let  now  y  =  Fx,  and  Y  =  F  (x  +  h),  and  assume,  agreeably  to 
art.  (4) 

Y  =  y  +  A/i  +  B/t^  +  C/i^  +  &c. 

A,  B,  C,  &c.  being  unknown  functions  of  x,  which  it  is  now  required 

to  determine. 

Suppose,  first,  h  to  be  variable  and  x  constant,  then,  differentiating 

on  that  supposition,  we  have 

^Y 

^  =  A  +  2B/i  +  ZCh^  +  &c. 
an 

Suppose,  secondly,  that  x  is  variable  and  h  constant,  then  the  dif- 
ferential coefficient  is 

dY       dy   ,    dA  ,    ,dB  .,dC   ,^,    ^ 
dz        dx        dx  dx  dx 

But  by  the  lemma  these  two  differential  coefficients  are  identical, 
hence  equating  the  coefficients  of  the  like  powers  of  A.  there  results 


that  is 


.   _  dy        _  dA_    „  _  <iB 
dx  2dx'  3dx' 


._%    B-^    -    C=^     -i-&c 
^~Tx'  ^~  dx^-2'^        dx^'2-3'  ^*'- 


Hence  the  required  development  is 

■    dy    h   ,    d^y         h^     ,   d^y  h^        ,   , 

If /i  is  negative,  the  signs  of  the  alternate  terms  will  be  negative. 

When  we  wish  for  the  development  of  the  function  Y  =  F(x  +  ^) 
in  any  particular  state,  that  is  when  x  takes  a  given  value,  Ave  have 
only  to  substitute  this  value  for  x  in  the  general  expressions  for  the 
coefficients  previously  determined,  and  we  shall  have  the  develop- 
ment according  to  the  above  form,  that  is,  provided  of  course,  that 
the  development  in  such  form  is  possible.  But  if  the  value  chosen 
for  X  render  the  development  impossible,  the  impossibility  will  be 
intimated  to  us  from  the  circumstance  of  some  of  the  terms  becoming 
infinite,  as  explained  in  art.  (4).  It  may,  however,  be  proper  here 
to  remark,  that  even  in  these  cases  of  impossibility,  the  leading  terms 
of  the  development  as  given  by  Taylor's  theorem,  are  still  true  as  far 
as  the  first  term  that  becomes  infinite.     But  as  we  propose  to  devote 

0 


42  THE  DIFFERENTIAL  CALCULUS. 

hereafter  an  entire  chapter  to  the  examination  o{  the  failing  cases  of 
Taylor's  theorem,  we  shall  not  enter  into  the  inquiry  here. 

(31.)The  theorem  of  Maclaurin  may  be  easily  deduced  from  that 
of  Taylor,  thus : 

Let  X  take  the  particular  value  x  =^  0,  then 

„,        r-r^  -T    ,    r<^Fa7^/i    ,   -drF X .    h?      ,   ^(PFx^       h? 

[Y]  =  F^  =  [F.J  +  [-^]  y  +  [-^Irr^  +  [^]  rr^Ts 

+  &c. 
Now  each  of  these  coefficients  is  constant,  and  therefore  indepen- 
dent of  the  value  of  h,  hence  h  may  take  any  value  whatever,  without 
affecting  these  coefficients;  we  may  therefore  call  it  x,  it  being 
observed  that  although  x  appears  in  the  notation  of  the  coefficients, 
it  does  not  appear  in  the  coefficients  themselves.  It  follows,  there- 
fore, that 

which  is  Maclaurin's  theorem,  before  investigated. 

EXAMPLES. 

(32.)     1.  To  develop  sin.  {x  -f-  ^)  in  a  series  of  powers  of  the 

dii  dhi  .        d^y 

arc  h.    Let  y  =  sm.  ^  •'•  ^  =  cos.  x,—  =  —  sm.x,^  =  — 

cos.  x,  Sec.  hence,  by  Taylor's  theorem, 

sin.  (x  -\-  h)  =  sm.  x  +  cos.  x  h  —  sm.  x- — -  —  cos.  x  - — -— - 

+  &c. 

+  COS.  X  (/t  -  3-:^  +  ^P^^^  -  &c.) 

The  series  within  the  parentheses  are  respectively  equal  to  cos.  h 

and  sin.  h  (p.  30,)  hence  the  property 

sin.  (a;  -{-  ^)  =  sin.  x  cos.  h  +  sin.  h  cos.  x, 

2.  To  develop  cos.  {x  -\-  h). 

dy  .         d^y  d^y 

y  =  cos.  X  .'.  -^  =  —  sm.  re,  j^  =  —  cos.  x,  ^  =  sm.  «,  &c. 


THE  DIFFERENTIAL  CALCULUS.  43 

Hence 

cos,  (x  +  A)  =  COS.  X  —  sin.  xh  —  cos.  x     ^   ■  +  sin.  x 

+  &c. 

=  COS.  X  f  1 r: &C.) 

^  1.2        1-2-3-4  ' 

—  sin.  X  (/4  —  - — - — r^  +  - — ; —  CSC. 

^         1 •2- 3       1 •2-a-4-o 

Hence  the  property* 

COS.  {x  +  h)  =  COS.  X  cos.  h  —  sin.  x  sin.  h. 
From  this  property,  and  the  analogous  one  for  the  sine  of  x  +  ft 
deduced  in  last  Example,  the  whole  theory  of  trigonometry  flows. 
By  putting  h  =  (m  —  1)  x,  these  two  properties  become 

sin.  mx  =  sin.  x  cos.  (m  —  1)  ^  +  sin.  (m  —  \)  x  cof.  x. 
cos.  mx  =  cos.  x  COS.  (m  —  \)  x  —  sin.  x  sin  (m  —  1)  x. 
Two  equations  which  will  be  employed  to  abridge  the  expressions  for 
the  diiferential  coefficients  in  the  next  Exam^ilc. 
3.  To  develop  tan.  ~  {x  +  h). 
y    =  tdn.~*x 

— ^  = —  —  cos.  ^y. 

dx        sec.  -y 

^y         «  •  '^y         •    o        2 

— ^  =  —  2  sm.  y  cos.  y  -f-  =  —  sir.  2  y  cos.  ^y. 

KjLXT  (tx 

J_  =  —  2  (cos.  2y  cos.  ^i/  —  sm.  2y  sin.  y  cos.  y)-^ 

=  —  cos.  By  cos.  y  -p 
=  —  COS.  3y  COS.  ^y 
— ^  =  2-3  (sin.  3y  cos.  'y  +  cos.  3y  sin.  y  cos.  ^y)  -^^ 

=  2*3  sin.  47/  cos.  ^y  -r^  =  2  •  3  sin.  4y  cos.  "y 

U/X 

*  It  should  not  be  concealed  from  the  student  that  the  property  here  deduced  is, 
in  fact,  involved  in  that  which  we  have  employed  to  obtain  the  differential  of  a 
sine.  If,  however,  we  consider  this  differential  deduced  as  in  Note  A  at  the  end, 
then,  the  inference  above,  fairly  establishes  the  property  in  question. 


44  THE    DIFFERENTIAL   CALCULITS. 

——  =  2  •  3  •  4  (cos.  4y  cos.  *7j  —  sin.  4y  sin.  y  cos.  ^y)  —- 

=  2  •  3  •  4  cos.  5y  cos.  ^y-~  =  2  '  3  •  4  cos.  5y  cos.  't^ 

&c.  &c. 

hence 

._,,,,.     ,     „  ,   sin.  2m  cos.  ^w  , 
tan.-'  (a?  +  A,)  =  t/  +  cos.  "y  h ^  h^  — 

cos.  3wcos.  ^w  ,,  ,  sin.  4vcos.  %  ,,  ,  cos.  5wcos.  *«  ,. 

1 i  /i'  -I -J 1  h'  + ^ ^  h'  —  &c. 

o  4  5 

4.  To  develop  log.  (a;  +  k)  according  to  the  powers  of  h. 

log.(x  +  4)=.og.x+M(^_^+|,-^^)+&c. 

6.  To  develop  tan.  {x  +  h)  according  to  the  powers  of  A. 

tan.  {x  -\-  h)  =  tan.  x  +  sec.  ^x  .  ^  +  2  sec.  ^x  tan.  »  - — -  + 

1  *  4a 

2  sec.^  a?  (1  +  3  tan.^x)  +  &c. 

1  .  ^  .  o 

(33.)  By  means  of  the  theorem  of  Taylor  may  be  obtained  a  very 
commodious  and  useful  form  for  the  representation  and  subsequent 
determination  of  the  differential  coefficient,  when  the  function  is 
complicated,  thus : 

Let  u  =■  Fy,  y  being  any  function  of  x,  which  we  may  represent 

by  «  =  fx,  and  let  it  be  required  to  find  the  expression  for  -r-.     Let 

X  take  the  increment  h,  then  since  y  =fx,  the  corresponding  incre- 
ment of  y  will,  by  Taylor's  theorem,  be 

dx      ^  dx^  1-  2^  dx'  l-2-3^'^''' 

Call  this  increment  fc,  then  the  corresponding  increment  of  u  will  be 
^         du  ,    ,    dhi         Jc'      ,    „ 

that  is,  restoring  the  value  of  k 

^        du  .dy  ,    ,    dFy     h?      .    „      ^ 

dy^  ^dx     ^  dar  1  •  2  ^  "''"J 
+  &c. 


THE    DIFFERENTIAL   CALCULUS.  45 

Hence  dividing  by  the  increment  h  of  the  independent  variable, 
and  taking  the  limit  as  usual,  we  have 

dFy  _  du  _  du  dy       ,    _du    dy 
dx         dx       dy'dx'  '  dy'  dx 

It  appears,  therefore,  that  the  differential  coefficient  —is  found  by 

differentiating  the  function,  on  the  hypothesis  that  y  is  the  independ- 
ent variable,  and  then  multiplying  the  coefficient  thus  obtained,  by 
that  derived  from  y  considered  as  a  function  of  x. 

(34.)  The  following  examples  will  suffice  to  illustrate  this  mode 
of  finding  the  differential  coefficient. 

1 .  Let  u  =  ay  where  y  =  b". 

du  .  ^    ,   dy       .    .       , 

1st.  —  =  ay  log.  o,  2nd.  —  =  b'  log.  6, 
dy  cix 

du       du    du  w  7 , 1  1       I 

.-.  -r-  =  -r-  .  -p-  =  a*   6'  log.  a  log.  b. 
dx       dy    dx 

2.  Let  u  =  log.  y,  where  y  =  log.  x, 

du  _  1   dy  _  1 
'  '  dy       t/'  dx       X, 

du du    dy  1 

'da;       dy  '  dx       x  log.  x 

3.  LetM  — sin.  (     ^     y. 

a  -\-  X 

^     .     X      2                 du       d  sin.  y                  dy            2ax 
Put  ( — -— )    =y  .'.  —  — ^  -"°  "  ^  — 


a  -i-  x'         ^  '      dy  dy  "^  dx       {a  -\-  xf 

du  ,     X     ..  2ax 

-J-  =  cos.  ( — j — y .  - — J- — -5. 

dx  a  -jr  X        (a  +  xy 


4.  Let  u  =  cot.  a",  y  being  =  log. 


X 


Va"  +  3? 


du  2    „      „i  dy 

=  —  cosec.^  a" .  a^  log.  a,    ''  — 


'  '  dy  '       '  dx    ,  x{a^-\-aP') 

du  n  ,  a? 

.:  -J-  =  —  cosec.^  a"  .  a"  log.  a  .     ,  ,  - — —. 
dx  x[a-  +  X-) 

(35.)  Let  us  now  take  the  more  general  function  u  =  F  {p,  q), 

p  and  q  being  functions  of  a?,  and  suppose  that  when  x  becomes  x-{- 

h,  p  and  q  become  p  -\-  k  and  q  +  k'.     Call  this  latter  q',  then  in 

consequence  of  the  proposed  change  in  the  independent  variable,  the 

function  will  become  F  (g',  p  -\-  k),in  which  it  is  to  be  observed  q' 


46  THE  DIFFERENTIAL  CALCULUS. 

enters  as  if  it  were  a  constant,  since  it  is  unaffected  by  the  increment 
k  of  the  variable  p. 

Hence  by  Taylor's  theorem 

F  O  +  .)  =  F  (,  „)  +  '^lli^.  +  ^I^  ^ 

+  &c (1). 

But  if  in  F  {q',  p)  we  substitute  for  q'  its  equal  q  -\-  k'  we  have 

F  (g'»  p)=--Fip,q  +  k')=^u  +  -k'-\-—  JL^+  &c. . . .  (2) 

du 

,.lI(2^£)  =  *  +  ^,.+  &e....(3), 
dp  dp  dp  ^  ' 

and  thus  by  continuing  the  differentiation  may  all  the  coefficients  in 
(1)  be  developed  according  to  the  powers  of  k',  but  this  first  will  be 
sufficient  for  our  purpose. 

Substitute  in  the  first  two  terms  of  (1)  their  developments  (2),  (3) 
and  we  have 

flit  nil 

¥{q-\-k\p^k)  =M  +  — A:'+  &c.  +liA;  +  &c (4) 

But  A;  being  the  increment  of  the  function  p,  arising  from  x  taking 
the  increment  h,  and  k'  being  the  increment  which  the  function  q  takes 
from  the  same  cause,  it  follows  that 

dp.    ,    d'p    h?  ^9  7    1    ^^^9    ^^      I    c 

Hence  by  substitution  in  (4)  we  have  finally 

■^  ,      ,    1.         .    7\  ,    cdu     da        du     dp.  ,     ,    . 

F(j  +  fc,p  +  fc)=«+S^.^  +  g^.£?&  +  &0. 

du  _  du     dq       du     dp 
'  '  dx       dq'  dx       dp'  dx 
(36.)  Again:  let  there  be  three  functions  of  x,  viz.  «  =  F  (j7,  gr,  r) 
then  when  x  becomes  a:  +  A,  let  p,  q,  r  become  p  +  k,  q -{-  k' ,  r  -{■ 
k"  respectively,  and  put  r'  for  the  latter,  then  in  the  function  F  {p  -\- 
fc,  g  +  k',  r'),  r'  enters  as  a  constant,  hence  as  above 

,,     ,x  .    ,du     dq    ,    du     dp.  ,    ,    ^ 

where  u  =  F  (p,  q,  r')  .-.  putting  r  +  k"  forr'. 


THE    DIFFERENTIAL    CALCULUS.  47 

dv 

,    dv  ^,,    ,    ,,        du       dv   ,        '  dr 

u  =  v  +  —k"  +  &c.,— =  — + j-k   +  &c. 

dr  dq       dq  dq 

dv 

du       dv    ,        '  dr  ,„   ,    o 

=       -I- k'  +  &c. 

dp       dp  dp 

But  fc"  =  -7-h  -{■  &c.  consequently 
ax 

■V  r      17        17/       1  7//N  I   (^^^     ^^    r    ^"     dq      dv     dp 

+  &c. 
dv  _  dv     dr        dv     dq        dv     dp 

dx       dr     dx       dq     dx        dp  '  dx' 

and  so  on  for  any  number  of  functions.  Hence  the  rule  is  to  differ- 
entiate the  expression  with  regard  to  each  of  its  constituent  functions 
severally,  as,  if  all  the  ethers  were  constants,  their  sum  loill  be  the  re- 
quired differential. 

Cor.  If  p  is  simply  x  then  in  the  function  u  =  F(ar,  g), 

du du       du    dq 

dx       dx       dq  '  dx* 
and  in  the  function  w  =  F  (x,  q,  r), 

du  _  du       du    dq    ^^  du    dr 

dx       dx       dq     dx       dr     dx 

(37.)  We  must  not  confound  here  the—  on  the  left,  with  that  on 

the  right,  in  these  equations,  for  the  former  denotes  the  total  differen- 
tial coefficient,  of  which  the  latter  forms  but  a  part,  and  is  therefore 
called  a  partial  differential  coefficient.  It  is  to  be  regretted,  how- 
ever, that  analysts  are  not  agreed  as  to  the  best  means  of  distinguish- 
ing total  from  partial  differential  coefficients,  and  accordingly  in  most 
works  on  the  calculus  the  same  symbol  is  applied  indiscriminately  to 
both ;  a  circumstance  likely  to  prove  a  frequent  source  of  perplexity 
to  the  learner ;  and  to  avoid  which  we  shall,  throughout  this  volume, 
always  distinguish  the  total  differential  coefficient  by  enclosing  it  in 
braces,  so  that  the  two  equations  above  will  be  written  thus : 

,du^  du       du    dq 

dx  dx       dq  '  dx 


48  THE  DIFFERENTIAL  CALCULUS. 

.du.  _  du       du    dq       du    dr 
dx         dx       dq  '  dx       dr  '  dx' 
(38.)  We  shall  now  add  a  few  examples,  showing  the  apphcation 
of  the  rule  deduced  in  last  article. 

EXAMPLES. 

1.  Let  u  —  cot.  a*  .-.  {-j-l    =—-  +  __.  _i.     Now 
ax  dx       dy     dx 

dxi  „       dx^ 

-r-  =  —  cosec.-*  x^  —J— 
dx  dx 


—  =  —  cosec.^  x"  — T—  =  —  cosec.^  a"  yx^^ 


du    dy  ,   ,  dx''  ^  ,  dv 

-T-  .  -p-  =  —  cosec."'  x^  — —  =  —  cosec.^  x"  .  x^  log.  x  -^ 
dy    dx  dx  °      dx 

...S*i=_^eosec.=  ^(^+l„g..|). 

a  (x  +  y/x+l/x  +  i/x) 


Vx'+{x+Vx  +  ^x+V  xf 
Put  X  +  V  x  +  y  X  +  y  X  =  q  .:  u  =  F  {x,  q),  and 
du aqx 

du    dq  ax'  rfo  do  1  1  i 

dq     dx       (^^.^.)f     dx  dx  2^x      3^f^4^r 

hence 

A  =  "^-"g^  (n-l_  +  _L  +  J_). 

{X'  +    9=)  2  2X2  3a;3  4a;4- 

3.  Let  M  =  log.  tan.  -,  y  being  a  function  of  x. 

J  ,       X  X     dx 

J  a  tan.  -       sec.^  -  .  —  , 

f^^^  ^ y  ^ y   y  ^        dx 

dx 


du  , 


hence 


tan.  -                tan.  - 

y                y 

X                    X    xdy 
dtan.  -            sec.'-*— j- 

y              y    y" 

y 

sin. 

X                X 

-  cos.  - 

y       y 

xdy 

X                                              X 

tan.  -                       tan.  - 

y                   y 

f 

X 

sin.  -  cos. 

y 

X 

y 

THE  DIFFKKENTIAL  CALCULUS.  49 

_     ^ 

.du     _  ^  dx 

dx^  .     X  X 

v'sin.  -  COS.  - 

^      y       y 


4.  Let  u  =  log.  (x  —  a  +  >/  x^  —  '2ax) 

...  i!^!  =         ' 


6.  LetM  =  (cos.  x)  •'"■' 

cdu^        ,           V   ,       ,            ,                         sin.  'x 
•*»  i-T-i  =  (cos.  x)  "^  •  (cos.  X  log.  COS.  X ;. 

Implicit  Functions. 

(39.)  Hitherto  we  have  considered  explicit  functions  only,  or  those 
whose  forms  are  supposed  to  be  given.  We  ■shall  now  consider 
implicit  functions,  or  those  in  which  the  relation  between  the  indepen- 
dent variable  x,  and  function  y,  is  implied  in  an  equation  between  the 
two.,  and  which  may  be  generally  expressed  by 
«  =  F  (x,  y)  =  0. 

The  deductions  in  article   (36)   will  enable  us  very  readily  to 

find  the  coefficient  -p  from  such  equations,  without  being  under  the 

necessity  of  solving  them,  a  thing  indeed  often  impossible. 

If  we  turn  to  the  corollary  in  the  article  just  referred  to,  and  sub- 
stitute y  for  q,  we  find 

.du.  _^du       du    dy 

*dx'       dx       dy  '  dx 

But  here  «  =  F  (x,  y)  =  0,  therefore  \—\  =  0,  for  u'  —  u  being 

always  0, — - —  is  always  0 ;  hence, 

du       du     dy 
dx       dy     dx 

from  which  equation  the  differential  coefficient  is  immediately  deter- 
minable :  it  is 

dy  du    .    du 

dx  dx   '   dy'' 

7 


60  THE  DIFFEREJITIAIi  CALCtLUS, 

hence,  having  transposed  the  terms  all  to  one  side  of  the  equation,  we 
mxist  differentiate  the  expression  as  if  y  were  a  constant,  and  then 
divide  the  restdting  coefficient,  taken  with  a  contrary  sign,  by  that 
derived  from  the  same  expression,  on  the  supposition  that  x  is  a  con~ 
s'anl. 

EXAMPLES. 

1 .  Let  u  =  y^  —  2mxy  -{-  x''  —  a  =  0. 

da  du       „  dy       my  —  x 

—  2my  —  2x,—  =  2y  —  2mx   •     -^  —     ^ 


dx  dy  dx       y  —  mx 

2.  Let  M  =  a:'  +  daxy  +  7/^  =  0. 

dx  dy  dx  ax-\-  y^ 

If  the  second  differential  coefficient  be  required,  we  have 

dy  dy 

^,y  ^       {ax  +  f)  {2x  +  «  ^)  +  {x'  +  ay)  (a  +  2y  -£) 

dx^  {ax  +  t/^)2 

or  substituting  for  -~  its  value  just  found 

_         2aT/*  +  6aaPy^  +  2'x'^y  —  2a^xy 

~{^TW 

_         2xy  {y^  +  3axy  -\-  jP)  —  2a^xy 

_____  , 

that  is,  since  aP  +  Zaxy  -\-  tf  =  0, 

d^y   __      2a'^  xy 


dx^        {ax  +  y^Y 

3.  Let  my^  —  xy  =  m  to  develop  y,  according  to  the  ascending 

powers  of  X, 

du  du        ^      „  dy  y 

;-  =  7/,  1-  =  Zinf  —  X  .'.  -^  =  - — ^ 

dx  dy  dx       Zmy^  —  x 

therefore,  calling  the  successive  differential  coefficients  p,  q,  r,  &c. 

y  —  Smx^p  —  xp 

^  {3my^^  xf     ' 


THE  DIFFERENTIAL  CALCULUS.  5l 

_       3nM/<jf  +  2  •  3  myp-  +  xq       {y  —  3my''p  —  xp)  {12myp  —  2 ) 
{Snvf  —  xf  (3my'  —  xf 

&c.  &c. 

Therefore,  by  Maclaurin's  theorem 

^  ^  3m        Z'm' 

4.  Let  y^x  —  m^  (!/  +  ^)  =  0,  to  develop  y  according  to  the 
ascending  powers  of  x. 

Representing,  as  in  last  example,  the  successive  differential  coeffi- 
cients by  p,  q,  r,  &c.  we  have 

_  2  •  3  t/p2  2*3  m\2ijp  +  aj)^^)  _ 

Sa;!/'-^  —  ift^  (3-^1/^  —  '"^)^ 

2-3  p^  2-3.4  m7/  _       2  •  3  •  4 

'  '  ^  3x?/^  —  m^  {Zxy^  —  m^)-  m^ 

&c.  &c. 

Hence,  by  Maclaurin's  theorem, 

—  —  -^^    _  ^^'  _  & 

iM^  m* 

5.  Let  y^  -h  2xy -^  a^  ==  a\  to  find  -^ 

dx 


«.  Let^-^  -  2y  ^^1=-^  -  X  +  6  to  find  |/ 
(x  -*-  a)"*         "^      X  —  a  ax 

dy  _3x  —  26  —  a 
^  '  dx~    2v/x  —  6  * 

7.  Let  J/'  —  3y  +  X  =  0,  to  develop  y  according  to  the  as- 
cending powers  of  X, 

^        3        3*        3» 


52  THE  DIFFERENTIAI,  CALCULUS. 

8.  Let  mifx  —  y  =  m,  to  develop  y  according  to  the  ascending 
powers  of  ar. 

y  =  —  m  - —  m*x  —  3mV  —  &c. 

9.  Let  my'*  —  r'y  =  mx',  to  develop  y  in  a  series  o{ descending* 
powers  of  X. 

m*       3m'' 


OHAPTSB   V. 

ON  VANISHING  FRACTIONS. 

Fx 

(40.)  It  is  here  proposed  to  determine  the  value  of  a  fraction  -pr- 

in  the  case  in  which,  by  giving  a  particular  value  a  to  the  variable, 
both  numerator  and  denominator  vanish,  the  fraction  then  becoming 

Fa  _  0 

As  such  a  form  can  arise  only  from  the  circumstance  of  the  same 
factor  X  —  a  being  common  to  both  numerator  and  denominator,  it 
is  plain  that  if  we  can  by  any  means  eliminate  this  factor  before  our 
substitution  of  a  for  :r,  we  shall  then  obtain  the  true  value  of  the  fraction. 
Sometimes  the  vanishing  factor  is  manifest  at  sight,  and  may  be 
immediately  expunged,  as,  for  instance,  in  the  fractions 
{x  —  a)  X     x^  —  a^     a^  —  2ax  -{■  x^ 


hx  —  ab  ^  {a  —  a;)^' 


-,  &c. 


0 
each  of  which  becomes-  when  x  =  a,  and  obviously  contains 

the  factor  x  —  a  in  both  numerator  and  denominator.     In  these  cases, 


*  This  will  be  effected  by  substituting  —  for  i^,  which  will  transform  the  equa- 

z 

tion  into  my'z  —  y  =  m,  then  developing  y  according  to  the  ascending  powers  of 
3,  and  afterwards  restoring  the  value  of  ar. 


TUB    DIIFEKENTIAL    CALCULTTS.  53 

therefore,  we  at  once  see  that  the  values  of  the  fractions  when  x  =  a 
are,  severally, 

p  —  CD  ,  0,  &c. 

In  certain  other  cases  the  value,  although  not  so  easily  seen  as  in 
the  foregoing  instances,  may,  nevertheless,  be  soon  ascertained,  by 
performing  a  few  obvious  transformations  on  the  proposed  fractions. 
Take  the  following  example  : 

v/a?  —  \/  a  +  s/{x  —  a) 
\/x^  —  a^ 
which  becomes  ^  when  x  =  a. 
This  fraction  is  the  same  as 

\/x  —  s/  a  1 

+ 


\/x^  —  a^         \/x  +   a 
and  the  first  of  these  terms  is  the  same  as 


\/  X  —  a 


y/ {x' —  d'){^/ X  +  ^/ af         v/  (x  +  a)  (n/x+  V  a) 
and  this  when  x  =  a  is  =  0,  therefore  the  value  of  the  proposed 

1 
fraction  when  x  =  a  is    . — • 
y/2a 

(41.)  But  the  most  direct  and  general  method  of  proceeding  de- 
pends upon  the  differential  calculus,  and  upon  the  development  of 
functions,  and  the  principal  object  of  the  present  chapter  is  to  ex- 
plain this. 

We  shall  premise  the  following  lemma,  viz.  In  the  general  de- 
velopment 

F  (x-f /O  =  Fx  + -^/t  +  -^^-^  +  &c. 

it  is  impossible  that  any  particular  value  given  to  x  can  cause  Fx, 
and  at  the  same  time  all  the  differential  coefficients,  to  vanish. 

For  if  such  could  be  the  case,  then,  for  that  particular  value  c,  we 
should  have 

F  (a  +  4)  =  0, 
whatever  be  the  value  of  A;  but  Fa  =  0,  and  therefore  the  prece- 


54  THE  DIFFERENTIAL  CALCULUS. 

ding  equation  can  exist  only  when  h  =  0,  whereas  the  hypothesis 
supposes  it  to  exist  independently  of  the  value  of  h. 

Let  now,  in  the  proposed  fraction,  x  be  changed  into  x  -\-  h,  then, 
by  developing  both  numerator  and  denominator,  it  becomes 
Fx  +  ¥'xh  +  F"xh^  +  F"'xh^  +  &c. 
fx  -\-f'xh  -\-f"xh?  ■^f"'xK'-\-  &c.  *  *  •  •  ^^^' 
where  F'x,  F''x^  &ic,f'x,f''x.  &c.  are  put  for  the  successive  differ- 
ential coefficients  divided  by  as  many  of  the  factors  1  •  2  •  3,  &c.  as 
there  are  accents. 

If  in  this  we  substitute  a  for  x,  then,  since  both  Fx  and  fx  vanish, 
the  fraction  becomes,  after  dividing  numerator  and  denominator  by 
h, 

F'a  +  F"ah  +  F"'ah'  +  &c. 

fa  ^-f'ah  +f"'ah?  +  &c.    '    '    '    *    ^2)' 

Fa 
and  this  fraction  when  h  =  Q  must  obviously  be  equal  to  -p;— ,  that  is 

Fa  _  F'a 

If,  however,  both  F'a  and  f'a  are  also  0,  then,  expunging  these 
terms  from  the  fraction  (2)  and  dividing  numerator  and  denominator 
again  by  ft,  we  have,  when  A  =  0, 

Fa  _  F"a 

Fa  .  .  . 

aad  so  on,  till  we  at  length  obtain  for  -^  a  fraction  of  which  the  nu- 
merator and  denominator  do  not  both  vanish,  and  such  a  fraction  we 
eventually  shall  obtain  by  virtue  of  the  preceding  lemma. 

Hence  the  following  rule  to  determine  the  value  of  a  fraction  whose 
numerator  and  denominator  both  vanish  when  x  =  a,  viz.  For  the 
numerator  and  denominalor  substitute  their  first  differential  coefficients, 
their  second  differential  coefficients,  and  so  on  till  we  obtain  a  fraction 
in  which  numerator  and  denominator  do  not  both  vanish,  for  x  =  a, 
this  xoill  be  the  true  value  of  the  vanishing  fraction. 

EXAMPLES. 

o'  —  b'    , 
1.  Required  the  value  of when  x  =  0. 


THE  DIFFERENTIAL,  CALCULUS.  55 

F'x       ,  ,       ,     ,        F'a       ,  ,       ,       ,       « 

-77—  =  log.  a  .  a'  —  log.  b  .  b'  .'.  --^—  =  log.  a  —  log.  6  =  log.  r 
Jx  J  a  o 

jP 3a;  +  2 

2.  Required  the  value  of  — r „   ,    ^ when  x  =  1. 

X  —  oar  +  8x  —  o 

F'a;  3.t2  — 3  F'o 


/x        4a;^  — 12x4-8       /'a        " 
differentiating  again 

F"    F'x 6t  ^  F'^g  _  ,  _ 

~r"f^~    12x^—12  *'*  7^  ~  ^  ~  °°  ' 

3.  Required  the  value  of  — '- '—,  when  x  =  90°. 

sm.  X  +  cos.  X  —  1 

F'x  _        COS.  X  -f  sin.  x       F'g 

f'x  COS.  X  —  sin.  X  *     y'g 

4.  Required  the  value  of 

X  +  x"—  {n-\-\f  ap-^'  +  {2n'-\-  2»  —  1)  x"+''  —  n^x^' 

when  X  =  1, 

F'x_ 

1  +2x— (n+l)V  +  (?t+2)(2n^  +  2n— l)x"+'  —  n''  (n  +  3)x"+2 
______ 

F'g_  0 
'"'  A  ~  0 

F"x  _ 

7^~ 

2— n(n+iyx^^+(n+l)(2»^+6ji='+3n— 2)x"— (n+2)(n='+3ji=')x^^ 


6(1- 

-X) 

F"g 

_  0 
"  0 

F"'x 



f"'x 

— (ri»— w)  (Ti+l)3x"-''+(n«4-n)  (2n3+6n''+3n— 2)x"-'  (w+1 )  (n-f2)  (n»-f  3n»)a;" 

__ 

F'"g  _  n  (n  +  1)  (2n  +  1)  _  Fg 
•■•  7^  6  ""  Ja~- 


d6 


THE  DIFFERENTIAL  CALCULUS. 


6.  Required  the  value  of — when  a:  =  0. 

XT 

F"a  _    1 

X  sin*  X  — '  90*^ 
B.  Required  the  value  of when  x  =  90°. 

COS.  X 


1  —  X 

7.  Required  the  value  of when  x  =  1, 

t 

cot.  x  - 

2 

¥'a  _  2 

fa       <!(' 

8.  Required  the  value  of-; when  a;  =  a. 

log.  a  —  log.  X 

Ya 

(42.)  If,  in  the  application  of  the  foregoing  rule,  we  happen  to  ar- 
rive at  a  differential  coefficient,  which  becomes  infinite  for  the  pro- 
posed value  X  =  a,  we  must  conclude  that  the  development  accord- 
ing to  Taylor's  Theorem  is  impossible  for  that  particular  value  of  the 
variable  ;  and  that,  therefore,  the  rule  which  is  founded  on  the  possi- 
bility of  this  development  becomes  inapplicable.  The  process,  how- 
ever, to  be  adopted  in  such  cases  is  still  analogous  to  that  above* 
depending  upon  the  development  of  the  numerator  and  denominator 
of  the  proposed  fraction  ;  but  here  this  development  must  be  sought 
for  by  the  common  algebraical  methods. 

Fa? 
As  before,  let  —^r-  be  a  fraction  which  becomes  %  when  we  change, 

.r  into  a.  Substitute  a  -f-  /i  for  x,  and  let  the  terms  of  the  fraction  be 
developed  according  to  the  increasing  powers  of  /i,  either  by  involu- 
tion, the  extraction  of  roots,  or  some  other  algebraical  process,  thea 
we  shall  have 

F  (a  +  /t)       A/i""  +  B/i^  +  &c. 


/(a  +  /i)        v;.«'4-R'A,.^ 


A'/j"-  -fB'/i'    +&C. 


THE  DIFFEKKNTIAL  CALCULUS.  57 

a  and  a  being  the  smallest  exponents  in  each  series,  (3  and  /5'  the 
next  in  magnitude,  and  so  on.  Now  these  three  cases  present  them- 
selves, viz. 

1°.  a  >  a' ;  2°.  a  =  a' ;  3°.  a  <  a'. 

In  the  first  case  by  dividing  the  two  terms  of  the  fraction  by  h'^', 
and  then  supposing  h  =  0,  there  results 
Fa         0        ^ 

j^=v  =  '- 

In  the  second  case  the  result  of  the  same  process  is 
Fa  _  A 

jr~  A'- 

In  the  third  case,  by  dividing  the  two  terms  of  the  fraction  by  /t«» 
and  then  supposing  h  =  0,  the  result  is 

Fa        A 

>r  =  ^  =  "- 

It  appears  from  these  results  that  the  development  of  the  numera- 
tor and  denominator  need  not  be  carried  beyond  the  first  term,  or  that 
involvuig  the  lowest  exponent  of  h,*  and  according  as  the  exponent 
in  the  numerator  is  greater  than,  equal  to,  or  less  than  that  in  the  de- 
nominator, will  the  true  value  of  the  fraction  be  0,  finite,  or  infinite. 
We  have,  therefore,  the  following  rule  : 

Substitute  a  +  /i  for  x,  in  the  proposed  fraction.  Find  the  term 
containing  the  lowest  exponent  of  ^,  in  the  development  of  the  nu- 
merator, and  that  containing  the  lowest  exponent  of  h  in  the  develop- 
ment of  the  denominator.  If  the  former  exponent  be  greater  than 
this  latter,  the  true  value  of  the  fraction  will  be  0,  if  less,  it  will  be  in- 
finite. But  if  these  exponents  are  equal,  divide  the  coefllicient  of  the 
term  in  the  numerator  by  the  coefiicient  of  that  in  the  denominator, 
and  the  true  result  will  be  obtained. 

This  method,  which  is  applicable  in  all  cases,  may  frequently  be 
employed  advantageously,  even  where  the  preceding  rule  applies. 

EXAMPLES. 

(a:3  _  3«.r  -f  2a2)^ 
9.  Required  the  value  of , —  when  x  =  a. 

*  The  first  term  wliich  actually  appears  in  the  development  is  of  course  meant 
here.  Those  which  may  vanish  in  consequence  of  the  coefficient  vanishing  not 
being  considered. 

8 


08  THE   DIFFERENTIAL  CALCUIiUS, 

Substituting  a  +  h  for  x,  we  have 

F  (o  4-   h)  _  hi  {h  —  ay*  _   (—  ah)^  +  &c. 

/  (a  +   h)       ^^  ^3^3  _^  3^^  ^  j^^^l  (3a'hy  +  &c. 

Since  the  exponent  of  h  in  the  numerator  exceeds  that  in  the  de- 
nominator, we  have 

Fa 

/«  

10.  Required  the  value  of ~ — — —  when  x 

Vx'—a^ 

=  a  (see  p.  53.) 

Substituting  a  +  ^  for  x. 


F{a  +  h)  • 

(a-^h)^  —  a^  -{-h^ 

h^  4-  &c. 

/(«  +  /»)   ~ 

h^  (2a  +  h)^ 
Fa            1 

(2a/i)^  +  &c. 

11.  Required  the  value  of—- — , -^ when  x  =  a. 

^  (1  4-  X  —  ay  —  1 

Substituting  a  +  h  for  t. 

F  (a  4-  ^)         h^  {2a  +  h)'^'  -\-  h  h  -{-  Sac. 


f{a-\-h)  (1  +  A)^— 1  3/1 +&C. 

Fa   _ 
••>    -  ^• 

to    i>       •    ^*K       1        .a{4a'+4x^)^  —  ax  —  a'     , 

12.  Required  the  value  of when  r 

(2o=»  +  2x^)7  _  a  —  X 
=  a. 

Substituting  a  +  ii  for  x. 

F{a+h)         a  (8a'  +  l^a^'h  +  12a/i2  +  Ah'y  —  2a'  ~  afe 

/(o  +  /»)  (40==  +  40/1  +  2^=")^  —  2a  — /i 

♦  To  develop  this  according  to  the  ascending  powers  of  h  we  must  write  it 
thus:  (  —  a  -{•  A)*  and  apply  the  binomial  theorem  when  we  have  the  seriea 
(  _  fl)l  4-  *  a—i  h  4-  &c. 


THE  DIFFERENTIAL  CALCULUS.  59 


which, 

by 

actiKilly  extracting  the  roots  indicated, 

a  (2a  4-  A  + 

2a 

+  &c. 

—  2a- 

-/,,) 

2a  +  h-{- 

h? 
4a 

+  &c. 

—  2a 

—  h 

a(-+&c.) 

Fa    _ 

1 

T 

4^ 

*9a 

This  example  is  perhaps  more  easily  performed  by  diflerentiation, 
according  to  the  first  rule  :  thus 

F'x  __  g  (4a^  +  4^')  "^  4^"  —  a       F'a         0 
~fx  ~      {2d'  +  2x')  -^  2x  — T  •  ''fa    =   0 

T'x  _  —  a  (4a^  +  Ax")  "  ^  32 j'  +  a  (4a^  +  4x"')~3  Sx 

J"'  _  (20"  +  2r')  "^  4ar»  +  2  (20=*  +  2x^)~^ 

F"a    _  2  _ 
•■•  7^   -   1  -  2a. 

a 
(43.)  Having  thus  seen  how  to  determine  the  value  of  any  fraction 
of  which  the  numerator  and  denominator  become  each  0  for  particu- 
lar values  of  the  variable,  we  readily  perceive  how  the  value  may  be 
found  when  particular  substitutions  make  the  numerator  and  denomi- 

Fa        a, 
aator  each  infinite.     For  if  —^  =  ^  then  obviously 

1 
Fa  _  >^  _  ^ 

Fa 

So  that  if  we  find,  by  the  preceding  methods,  the  value  of  this  last 
fraction,  the  value  of  the  proposed  fraction  will  be  also  obtained. 
The  following  example  will  illustrate  this. 

1          X 
tan.  (-  *  .  -) 
^2         a' 
1 3.  Required  the  value  of    ,   _,  .^ ^rrr  when  .r  =  a. 


<»<>  THE  mPFEUENTlAL  CALCUHTS. 

In  this  case  the.  fraction  takes  the  form  ^ ,  therefore, 

CO 

1 

fi^  _  a  (.r=  —  a")  x~-       F'x    _  2a^  .r" 

1 : 


Fa:  cot.<^^.l)        ^'^  _  cosec^^  *  •  ^)  ^ 

Fa  _  ^  _       4a 

/'a  -rr    ~         * 

~  2^ 

(44.)  By  the  same  principles  we  may  also  find  the  true  value  of  a 

product  consisting  of  two  factors,  which  for  a  particular  value  of  the 

variable  becomes  the  one  0  and  the  other  cc.     For  if  Fa  =  0  and 

/a  =  CO  ,  then, 

„  ..         Fa        0 

Fa  X  fa  = =  - 

-^  1  0 

7^ 
We  shall  give  an  example  of  this. 

14.  Required  the  value  of  the  product  (1  —  x)  tan.  (|  -jex)  when 
a:  =  1, 

In  this  case  the  first  factor  becomes  0  and  the  second  co  . 

-  .-.  F'x  X  f'x  = 


1  1  cot.  (^  *  0.-)  -  2  /  ,        \* 

Jx  tan.  (i  ifx)  ^2       '  ^ 

.-.  F'a  X  fa=-^  =  -. 

(45.)  And  finally,  by  the  same  principles,  the  true  value  of  the  dif- 
ference of  two  functions  may  be  ascertained  in  the  case  where  the 
substitution  of  a  particular  value  for  the  variable  causes  each  of  them 
to  become  infinite. 

For  if  Fa  —  ao  and /a  =  co  ,  then 

1  1 

'fa~Fa_0 


Fa -fa 


1  0 


Fa  X  fa 
The  following  example  belongs  to  this  case : 


THE    mPFERENTIAI.    CAIXTJIiUS.  61 

15.  Required  the  true  valr.e  of  the  diflference  x  tan.  x  —  \'k 
sec.  X,  when  x  =  90^. 

1  1  1 1 

fx        Yx  ^  ir  sec.  x  x  tan.  x 


1 


Fx  X  fx  X  tan.  x    X    i  cr  sec.  x 

by  substituting for  sec.  x,  and  then  dividing  numerator  and 

denominator  by  - 


^  nc  X  tan.  X 

T^,  /.,         -T  cos.  X  +  sin.  X 

.-.  F  X  —  fx  — : .'.  Fa  — /a  —  —  1. 

''  —  sm.  X  -" 

It  should  be  remarked  that  in  this,  as  well  indeed  as  in  the  prece- 
ding cases,  the  transformation  requisite  to  reduce  the  expression  to 
the  form  f  in  many  instances  at  once  presents  itself  to  the  mind, 
when  of  course  it  will  be  necessary  to  recur  to  the  preceding  formu- 
las.    The  example  just  given  is  one  of  these  instances,  for  since 

sin.  X         ,  1         ,  , 

tan.  X  = ,  and  sec.  x  = ,  the  proposed  expression  at 

cos.  X  cos.  X 

once  reduces  to 

X  sin.  X  —  1  * 


COS.  X 

which  ia  the  required  form. 

(46.)  We  shall  terminate  this  chapter  with  a  few  miscellaneous 
examples  for  the  exercise  of  the  student. 

x" 1 

16.  Required  the  value  of ,  when  x  =  1. 

X  —  1 

^ns.  n. 

fij^  -4—  CLC^  — ^—  ^OiCor 

17.  Required  the  value  of  7—5 —. ; — r-^,  when  x  =  c. 

bxr  —  2bcx  +  or 

Ans.  r- 
0 

3r    ^^—  CLOl    ^~"  (VX  "i"    Cb 

18.  Required  the  value  of ,  when  x  =  a. 

or  —  a 

Ans.  0. 

*  This  is  obtained  by  multiplying  the  last  fraction  above  and  below  by  a;  tan. 

,           .  .       sin.  X  .                    ,      1      - 
X  X  Jt  sec.  X,  then  writing  for  tan.  r,  and for  sec.  x,  Ed. 


62  THE   DIFFERENTIAL  CALCULUS. 

19.  Required  the  value  of : ^ -,  when 

a  —  {ax^)  * 

16a 
X  —  a.  Ans.  — — < 

9 

1 X  ~j~  loff.  X 

20.  Required  the  value  of — ,  when  x  =  \. 

1  —{2x  —  xY^ 

Ana,  —  1. 
x' X 

2 1 .  Required  the  value  of -— ; ,  when  x  =  1 . 

1  —  X  -v  log.  X 

Ana.  —  2. 

_       .     ,  ,         ,        -tan.  T  —  sin.  T      , 

22.  Required  the  value  of ; ,  when  x  =  0. 

sin.  3r 

Ans.{. 
[  X  =  \ 
Ana.  f 

when  X  =  a. 

(x  —  a)t 

Ana.  2a^. 

X  1 

26.  Required  the  value  of -; ,  when  x  =  1. 

^  X  —  1         log.  X 

Ana.  |. 

ax  —  XT  , 

26.  Required  the  value  °f  ^4  _  ga^x  +  2ax- -  x^'  ^^""  '  =  "' 

Ana.  CO  . 

1  X 

27.  Required  the  value  of  ^ ,  when  x  =  1. 

^  log.  X       log.  X 

Ana.  —  1. 

28.  Required  the  value  of — — —  .  tan.  — ,  when  x  =  a. 

4 
Am. . 


X  IO£f«  X  —  ix  ~~"  1  J 

23.  Required  the  value  of — -^ — -r-r ^»  when  x  =  1. 

(x  —  1)  log.  X 

24.  Required  the  value  of  ^ '—,  when  x  —  a 


„  a  ( 1  —  x)       ,  , 

29,  Required  the  value  of  — V-r — ^^  ^'^®"  x  =  1. 
^  cot.  I  <g  X 


2a 

Ana.  — . 


g" e 

30,  Required  the  value  of : ,  when  x  =  0. 

^  X  —  sin.  X 


Ans.  1. 


THE  DIFFERENTIAL  CALCULUS.  63 


OHAFTER  VI. 

ON  THE  MAXIMA  AND  MINIMA  VALUES  OF  FUNC- 
TIONS OF  A  SINGLE  VARIABLE. 

*(47.)  In  any  function  y  =  Fx  let  the  independent  variable  take  b 
particular  value  x  =  a,  as  also  a  preceding  and  succeeding  value 
X  —  a  —  h  and  x  =  a  -\-  h,  then  the  corresponding  values  of  the 
function,  arranged  according  to  those  of  the  variable,  will  be 

and  if  a  be  such  that  for  any  finite  value  of  A,  however  small,  and  for 
all  intermediate  values  between  this  and  0,  the  middle  value  Fa 
exceeds  that  on  each  side,  the  value  x  =  aia  said  to  render  the  func- 
tion a  maximum;  but  if  the  middle  value  continue  less  than  that  on 
each  side  between  the  same  limits  of  h,  the  value  x  =  ais  said  to 
render  the  function  a  minimum  ;|  so  that  we  are  not  always  to  un- 
derstand by  the  expression,  Jnaarmwrn  value  of  a  function,  the  g-rea/esf 
value  such  function  can  possibly  take  the  term  being  of  more  com- 
prehensive meaning,  applying  to  every  state  of  the  function  which 
exceeds  its  immediately  preceding  and  succeeding  state.  In  like 
manner,  the  minimum  value  of  a  function  does  not  always  imply  the 
least  possible  value  of  such  function,  but  equally  characterizes  every 
state  of  the  function  which  is  less  than  its  immediately  preceding 
and  succeeding  state. 

Before  proceeding  to  the  general  method  of  determining  the  values 
of  T,  necessary  to  render  any  proposed  function  a  maximum  or  a 
minimum,  we  must  premise  this  lemma  : 

Ij  the  function  Y  {a  -\-  h)  be  developed  according  to  the  ascending 
powers  ofh,  a  value  so  small  may  be  given  to  h  that  any  proposed  term 
in  the  series  shall  exceed  the  sum  of  all  that  follow. 

*  The  maximum  value  of  any  function  is  that,  which  is  greater  than  those  which 
immediately  precede  and  follow  it ;  and  the  minimum  value  is  that,  which  is  less 
than  those  which  immediately  precede  and  follow  it.  It  is  proper  to  observe  that 
the  same  function  may  have  several  maxima  and  minima  values.  Ed. 

t  This  definition  of  a  maximum  and  a  minimum  is  but  a  slight  alteration  of 
that  given  by  Dr.  Lardner  in  his  Differential  Calculus,  p.  103. 


64  THE    DIPFJiRKNTIAL    CALCULUS. 

(48.)  Let  A/i    be  any  proposed  term  in  the  development,  and  let 

B/i  ,  Ch^,  &c.  be  those  which  follow,  each  exponent  being  greater 
than  that  which  precedes  it.  We  are  to  prove  that  h  may  be  taken 
so  small  that 

A/i"  >  /i^  (B  +  Ch^~^  4-  D//"^  +&C.) 
Putting  S  for  the  sum  of  the  series  within  the  parentheses,  it  is 

B-a 
obvious  that  h  may  be  taken  so  small  that  S/i  may  be  less  than 

any  proposed  quantity  A,  and  that  therefore  if  h'  be  such  a  value  we 

must  have 

Ah'"-  >  S/i'^ 

B  —  a  . 
which  establishes  the  proposition.     As  S/i  is  less  than  A  for 

h  =  h',  the  expression  continues  less  than  A  for  every  value  of  h 

less  than  h'. 

(49.)  Let  us  now  inquire  by  what  means  we  may  determine  those 

values  of  x  which  render  any  proposed  function  Fa;  a  maximum  or  a 

minimum.     In  order  to  do  this,  let  x  be  changed  into  x  ±:  h,  then 

by  Taylor's  theorem 

„  ,  ,  ^        ^  dy  ,     ,   drri      h^  (Py        h^ 

F(,±.)  =  F.±^A+^— ±^^-^3  + 

(te*    1  •  2  •  3  •  4  * 
Now  if  0?  =  a  render  the  proposed  function  a  maximum,  then  there 
exists  for  h  some  finite  value  h',  such  that  for  all  the  intermediate 
values  between  this  and  0  we  have 

Fa  >  F  («  ±  /t), 
and,  consequently, 

But  if  this  value  render  the  function  a  minimum,  then,  for  all  the  in- 
termediate values  of /i  between  h  =  h'  and  h  =  0,  we  have 

Fa  <  F  {a  ±  k) 
and,  consequently. 


THE  DIFFERENTIAL  CALCULUS. 


65 


It  has,  however,  been  proved  above,  that  a  value  may  be  given  to 
h  small  enough  to  render  the  first  term  in  each  of  the  series  (1)  and 
(2)  greater  than  the  sum  of  all  the  other  terms,  and  that  this  first  term 
will  continue  greater  for  all  other  values  of  k  between  this  small 
value  and  0,  so  that,  for  each  of  these  values  of  ft,  the  sign  belonging 
to  the  sum  of  the  whole  series  is  the  same  as  that  of  the  first  term  ; 
it  is  impossible,  therefore,  that  either  of  the  conditions  (1)  or  (2)  can 

exist  for  both  +  [-^]  ft  and—  [^]  ft,  unless  [-^]  =  0  ;  we  con- 
clude, therefore,  that  those  values  of  x  only  can  render  the  function 
a  maximum  or  minimum  which  fulfil  the  condition 

ax 
expunging,  therefore,  the  first  term  from  each  of  the  series,  (1),  (2), 
we  have,  in  the  case  of  a  maximum,  ilie  condition 

AiJL.  ±  [%-!^  +  &c zo . .  .  (3).* 

and  in  the  case  of  a  minimum,  • 

r^J!L.  +   [^]-^^  +  &c.  7  0  .  .  .  (4). 

Now  the  former  of  these  conditions  cannot  exist  for  any  of  the  values 
of  ft  between  ft  =  ft'  and  ft  =  0,  by  virtue  of  the  foregoing  principle, 

unless  [  j3-]  is  negative,  nor  can  the  latter  condition  exist  unless 

[— ^]  is  positive,  that  is,  supposing  that  these  coefficients  do  not 

vanish  from  the  series  (3)  and  (4). 

We  may  infer,  therefore,  that  of  the  values  of  x  which  satisfy  the 

dti 
condition  ;p  =  0,  those  among  them  that  also  satisfy  the  condition 

— ^  Z  0  belong  to  maximum  values  of  the  function,  while  those  ful- 
dxr 

dj^y 
filling  the  condition  -j^  y  0  belong  to  mmimum  values  of  the  func- 
tion.    It  is  possible,  however,  that  some  of  the  values  derived  from 
the  equation  -t"  ~  ^  "^^.y,  when  substituted  for  x  in  -j-^,  cause  this 

*  See  Note  (C). 
9 


66  THE  DIFFERENTIAL  CALCULUS. 

coefficient  to  vanish,  in  which  case  the  conditions  (1),  (2),  become 
and 

which  are  both  impossible  unless  [  j-^]  =  0,  for  reasons  similar  to 

those  assigned  above,  and,  unless,  also  [-7-^]  /  0  in  the  case  of  a 

maximum,  and  f^-^l  7  0  w  the  case  of  a  minimum ;  that  is,  on  the 
ax* 

supposition  that  this  coefficient  does  not  vanish  from  the  series  (5) 

and  (6).     If,  however,  this  coefficient  does  vanish,  then,  for  reasons 

similar  to  those  assigned  in  the  preceding  cases,  the  following  coeffi- 

*^'6nt  j-j  must  also  vanish,  and  the  condition  of  maximum  will  then 

d'y  d?y 

^^  L T~r]  Z  0»  and  the  condition  of  minimum  [jtt]  7  0,  and  so  on. 

It  hence  appears,  that  to  determine  what  values  of  x  correspond 

to  the  maxima   and  minima  values  of  the  function  y  =  Fx,  we 

must  proceed  as  follows  : 

dy 
Determine  the  real  roots  of  the  equation  -p  =  0,  and  substitute 

them  one  by  one  in  the  following  coefficients  -7^,  -A^,  &c.  stopping 

at  the  first,  which  does  not  vanish.     If  this  is  of  an  odd  order,  the 

root  that  we  have  employed  is  not  one  of  those  values  of  x  that 

renders  the  function  either  a  maximum  or  a  minimum  ;  but  if  it  is  of 

an  even  order,  then,  according  as  it  is  negative  or  positive,  will  the 

root  employed  correspond  to  a  maximum  or  to  a  minimum  value  of 

the  function. 

(50.)  It  must  however  be  remarked,  that,  should  any  of  the  roots 

dy 
of  the  equation -1- =  0  cause  the  first  of  the  following  coefficients, 

which  does  not  vanish,  to  become  infinite,  we  cannot  apply  to  such 
roots  the  foregoing  tests  for  distinguishing  the  maxima  from  the 


THE  DIFFERENTIAL  CALCULUS.  67 

minima,  because  the  true  development  of  the  function  for  any  such 
value  of  a:  begins  to  differ  in  form  from  Taylor's  development,  at  that 
term  which  is  thus  rendered  infinite  (4),  so  that  we  cannot  infer, 
from  Taylor's  series,  whether  the  power  of /»,  which  ought  to  enter 
this  is  odd  or  even. 

In  a  case  of  this  kind,  therefore,  we  must  find,  by  actual  involu- 
tion, extraction,  &c.  the  true  term  that  ought  to  supply  the  place  of 
that  rendered  infinite  in  Taylor's  series  for  x  =  a.  If  this  term  take 
an  odd  power  of  A,  or,  rather,  if  its  sign  change  with  the  sign  of  A, 
then  X  =  a  does  not  render  the  function  either  a  maximum  or  a  mini- 
mum ;  but  if  the  sign  does  not  change  with  that  of /»,  then  the  value 
of  a:  renders  the  function  a  maximum  or  a  minimum,  according  6is 
the  sign  of  this  term  is  negative  or  positive. 

To  illustrate  this  case,  suppose  the  function  were 

y  =  b  -{-  {x  —  a)* 
..-^-3(x-«)3 
dhj    _  10  _  I 

dy 
Now  the  equation  T"  =  0  gives  x  =  a,  so  that  if  any  value  of  x 

could  render  the  proposed  function  a  maximum  or  a  minimum, 
this  most  likely  would  be  it.     By  substituting  this  value  of  x  in 

■j-j  the  result  is  infinite,  and  we  cannot  infer  the  state  of  the  function 

from  this  coefficient ;  therefore,  substituting  a  ±  /i  for  x  in  the  pro- 
posed, we  have 

F  (a  it  ;i)  =  6  ±  A* 

and,  as  h^  obviously  changes  its  sign  when  h  does,  we  conclude  that 
the  function  proposed  admits  of  neither  a  maximum  nor  a  minimum 
value. 
Again,  let 

y  —   h  -{■  {x  —  ay 
dy    _  4  I 


68  THB    DIFFERENTIAL    CALCULUS. 

_,  .      dy  ,         ,  .  ,  <^y    ^ 

The  equation  -^  =  0  gives  x  =  a,  a.  value  which  causes  -r-j-  to 

become  infinite  ;  therefore,  substituting  a  ±  /i  for  a:  in  the  proposed, 
we  have 

F  (o  ±  /i)  =  6  =  /i3 

.  i  .         .  . 

and,  as  the  sign  of /i=*  is  positive  whatever  be  the  sign  of  A,,  we  con- 
clude that  the  value  x  =  a  renders  the  function  a  minimum. 

(51.)  There  remains  to  be  considered  one  more  case  to  which  the 
general  rule  is  not  applicable,  and  which,  like  the  preceding,  arises 
from  the  failure  of  Taylor's  theorem.  We  have  hitherto  examined 
only  those  values  of  x  for  which  Taylor's  deyelopment  is  possible,  as 
far  at  least  as  the  first  power  of /(,  but  we  cannot  say  that  among 
those  values  of  x,  which  would  render  the  coefficient  of  this  first  power 
iiifinite,  there  may  not  be  some  which  cause  the  function  to  fulfil  the 
conditions  of  maxima  or  minima ;  therefore,  before  we  can  conclude 

dy 
in  any  case  that  the  values  of  x,  deduced  from  the  condition  j^  —  0, 

comprise  among  them  all  those  which  can  render  the  function  a 

maximum  or  minimum,  we  must  examine  those  values  of  ^arising  from 

dy 
the  condition  -y-  =  co  by  substituting  each  of  these  ±  hfor  x  in  the 

proposed  equation,  and  observing  which  of  the  results  agree  with  the 

conditions  of  maxima  and  minima  in  (47). 

(52.)  If  the  function  that  y  is  of  x  be  implicitly  given,  that  is,  if 

u  —  ¥  {x,y)  =  0; 

then,  by  (39),  we  have,  for  the  differential  coefficient, 

dy  du    .   du 

dx  dx       dy  '  '  '  ^  ^* 

dy  du 

and  therefore,  when  -7-  =  0,  we  must  have  j~  =^   0  5    hence,  the 

values  corresponding  to  maxima  and  minima,  are  determinable  from 
the  two  equations* 

dy  .  du 

*  Other  values  may  be  implied  in  the  condition      -  =  co ,  which  leads  to  — 

=  0,  but  to  ascertain  wliich  of  these  are  applicable  would  require  us  to  solve  the 
equation  for  y. 


THE    DIFFERENTIAL    CALCULUS.  69 

^  =  o\  '    '    •  (2)- 

dx  * 

Having  found  from  these  values  of  a:  that  may  render  y  a  maximum 

or  a  minimum,*  as  also  the  corresponding  values  oft/  itself,  we  must 

dry 
substitute  them  for  x  and  y  in  j-j,  when  those  values  of  y  will  be 

maxima  that  render  this  coefficient  negative,  and  Ihose  will  be  mini- 
ma that  render  it  positive.  But  those  values  that  cause  it  to  vanish, 
belong  neither  to  maxima  nor  to  minima,  unless  the  same  values 

,     d'y 
Qause  also  -7^-  to  vanish,  and  so  on. 
dx-^ 

The  second  differential  coefficient  may  be  readily  derived  from 

(1),  for,  putting  for  brevity 

we  have 

^  4-  ^    ^^  —  M  (—  4-  ^    ^^ 
d'y   ^  dx         dy  '  dx  dx         dy   '  dx 

M 

which,  because  =^  =  0,  becomes  for  the  particular  values  of  x  re- 
sulting from  this  condition, 

d'y     _         d'u     ^    du 

'-rf^-l"-L^J  •  Lrf^J    •    •    •  ^^■>' 

d^y 
By  differentiating  the  above  expression  for  -7-j  we  shall  find 

and  so  on. 

(63.)  Before  we  proceed  to  apply  the  foregoing  theory  to  exam- 
ples, we  shall  state  a  few  particulars  that  may,  in  many  instances,  be 
serviceable  in  abridging  the  process  of  finduag  maxima  and  minima. 

*  Gamier,  at  p.  271  of  his  CalcuL  Differential,  says,  that,  by  means  of  the 
equations  (2)  "  on  obtient  les  valeurs  de  x  et  1/  par  lesquelles  F  (x,  y)  devient  ou 
peut  devenir  maximum  ou  minimum  ;"  but  this  is  evidently  a  mistake,  since,  by 
hypothesia,  F  (x,  y)  is  always  =  0. 


70 


THE  DIFFERENTIAL  CALCULUS. 


1.  if  the  proposed  function  appears  with  a  constant  factor,  such 
factor  may  be  omitted.  Thus,  calling  the  function  Ay,  the  first  dif- 
ferential coeflicientwill  be  A  -^,  and  A -^  =  0  leads  to  -^=0,also 

ax  ax  dx 

— —  =  0  leads  to  ^  =  0,  so  that  A  may  be  expunged  from  the 

dx  dx 

function. 

2.  Whatever  value  of  x  renders  a  function  a  maximum  or  mini- 
mum, the  same  value  must  obviously  render  its  square,  cube,  and 
every  other  power,  a  maximum  or  minimum  ;  so  that  when  a  proposed 
function  is  under  a  radical,  this  may  be  removed.  The  rational 
function  may,  however,  become  a  maximum  or  a  minimum  for  more 
values  of  x  than  the  original  root ;  indeed,  all  values  of  x  which 
render  the  rational  function  negative  will  render  every  even  root  of  it 
imaginary ;  such  values,  therefore,  do  not  belong  to  that  root ;  more- 
over, if  the  rational  function  be  =  0,  when  a  maximum,  the  corres- 
ponding value  of  the  variable  will  be  inadmissible  in  any  even  root, 
because  the  contiguous  values  of  the  function  must  be  negative. 

3.  The  value  x  =  cb  can  never  belong  to  a  maximum  or  minimum, 
inasmuch  as  it  does  not  admit  of  both  a  preceding  and  succeeding 
value. 

EXAMPLES. 

(54.)     1.  To  determine  for  what  values  ofar  the  function 
y  ■=  a^  ■{■  ly'x  —  c^  x" 
becomes  a  maximum  or  minimum, 

dx  dx" 

From  the  second  equation  it  appears  that,  whatever  be  the  values  of 

dy 
X,  given  by  the  condition  -^  =  0,  they  must  all  belong  to  maxima. 

From  6^  —  2c^x  =  0,  we  get  x  =  — -j  ;  hence 

when  X  —  —-r-  .•.  r/  =  a*  +  -— r,  a  maximum. 

dy 
The  equation  -r-  =  cc  would  give,  in  the  present  case,  z  =  oo,  a 

value  which  is  inadmissible  (53). 


tion 


THE  DIPPBRENTIAJL  CALCULUS.  71 

2.  To  determine  the  maxima  and  minima  values  of  the  func- 
y  =  3oV  —  b'x  +  c* 


putting 


ax  ax' 


ga^x'  —  b'  =  0  .'.X  =  ±  — 
3a 


Substituting  each  of  these  values  in -^  we  infer  from  the  results 

that 

when  X  =  —  .  .  .  .  y  =  c ,  a  mm. 

3a  ^  9a 

b''  ,   ,    26" 

X  = .  .  •  .  y  =  c^  +  ,  a  max. 

3a  ^  9o 

3.  To  determine  the  maxima  and  minima  values  of  the  function 

y  =  A/2ax. 
Omitting  the  radical 

du 
u  =  2ax  .'.  -T-  =  2a, 
ax 

as  this  can  never  become  0  or  co  ,  we  infer  that  the  function  has  no 
maximum  or  minimum  value. 

4.  To  determine  the  maximum  and  minimum  values  of  the 
function 


y  =  \f  ^(^3?  —  2ax^. 
Omitting  the  radical  and  the  constant  factor  2a  (63), 
M  =  2ar'  —  a^, 

..._=4ax-3x»,^=4a-6a:, 

4a 
.*.  X  (4a  —  3a;)  =  0  .*.  x  =  0,  or  x  =  -^. 

Substituting  each  of  these  values  in  -j-r-,  the  results  are  4a  and 

ax* 

—  4a ;  hence 

when  X  =  0  .  .  .  y  =  0,  a  minimum. 


73  THE  DIFFERENTIAL  CALCULUS. 


_  4a  _  8 

^  -  y    •  •  2/  -3 

If,  instead  of  the  preceding,  the  example  had  been 


„     •  •  J/  —  o  «^  maximum. 


y  =  \/2aar*  —  4aV, 
we  should  have  had 


du  (Pu 

=  6x  —  4a. 

4a 


-T~  =  ^3^  —  4ax,  -j-^  =  6x  —  4a. 


X  (3x  —  4a)  =  0,  .'.  X  =  0,  or  X 


the  same  values  as  before  ;  but  the  first  corresponds  here  to  a  maxi- 
mum, since  it  makes  - —  negative  ;  this  value,  therefore,  must,  by 

(53),  be  rejected.     If,  indeed,  we  substitute  0  ±  ^  for  x,  in  the  pro- 
posed function,  it  becomes 


y  =  V—  4a^h'  =F  2a/i^ 
\yhere  h  may  be  taken  so  small  as  to  cause  the  expression  under  the 
radical  to  be  negative  for  all  values  of  h  between  this  and  0. 

5.  To  determine  the  maxima  and  minima  values  of  the  function 


y  =z  a-\-  \/d^  —  2a-x  +  ax^. 
If  t(  is  a  maximum  or  minimum,  y  -r-  a  will  be  so ;  therefore,  trans- 
posing the  a,  cuid  omitting  the  radical  (53), 
u  =  a^  —  2a^x  +  ax^ 

-J-  =  —  2a''  +  2ax,  r-—  =  2a, 
ax  air 

.:  —  2a^  +  2ax  =  0  .*.  x  =  a, 

.*.  when  X  =  a  .  .  .  y  =  a,  a  minimum. 

6.  To  determine  the  maxima  and  minima  values  of  the  function 


'  (a  -  x)^ 
In  solving  this  example  we  shall  employ  a  principle  that  is  often  found 
useful,  when  the  proposed  function  is  a  fraction  with  a  denominator 
more  complex  than  the  numerator.  Instead  of  the  function  itself  we 
shall  take  its  reciprocal,  which  will  give  us  a  more  simple  form,  and 
it  is  plain  that  the  maxima  and  minima  values  of  the  reciprocal  of  a 


THE  DIFFERENTIAL  CALCULUS.  78 

Function  correspond  respectively  to  the  minima  and  maxima  of  the 
function  itself.  Omitting,  then,  the  constant  a^,  and,  taking  the  re- 
ciprocal, we  have 

a^  —  2ax  +  x"       a?        ^      , 
u  = = 2a  +  X 

X  X 

'  '  dx  x^         '  dr^         x^ 

a"    ,  d-iL  .    2 

.-.  —  -V  +  1  =  0  .-.  X  =  ±  a  .-.  [  j— ]  =  ±  -, 

ar  aJT  a 

hence  x  =  a  makes  m  a  minimum,  and  z  =  —  a  makes  it  a  maxi- 
mum, therefore 

when  X  =  a  .  .  .  y  —  co ,  a  maximum, 

x  =  —  a  .  .  '  y  =^  —  i^j^  minimum. 

7.  To  determine  the  maxima  and  minima  values  of  the  function 


V  =b  +  1/  {x  —  ay. 
Omitting  b  and  the  radical 

u  =  (x  —  ay 

,,_=6(x-ar,— ==4.5(x-a/ 

.-.  5  (x  —  o)"  =  0 .-.  X  =  a  .-.  f-j-^]  =  0. 

As  this  coefficient  vanishes,  we  must  proceed  to  the  following, 
which  however  all  contain  x  —  a,  and  therefore  vanish,  till  we  come 

to  -r-r  =  2  •  3  •  4  •  5 ;  as  therefore  the  first  coefficient  which  does 
dxr 

not  vanish  is  of  an  odd  order,  the  function  does  not  admit  of  a  maxi- 
mum or  a  minimum. 

8    To  determine  the  maxima  and  minima  values  of  the  function 


dy  d?y  1 

■£  =  X'  (1  +  log.  ^).  5;^  =  ^  I-  ^-  (»  +  log.  ^Yl 

10 


74  THE  DIFFEUENTIAL  CALCULUS. 

The  factor  x'  can  never  become  0,  therefore 

(1  +  log.  x)  =^  0  .-.  log.  X  =  —  1. 

1 

.*.  a:  =  c~^  =  - 
e 

.   r^]    =    (!)■. . 

1 

.'.  when  X  =  — ,  x*  =  ( — )   ,  a  minimum. 

9.  To  determine  the  maxima  and  minima  values  of  y  in  the 
function 

u  =  aP  —  3axy  -\-  y^  =  0 

du 

-  =  3ar'-3ai/.-.  (52) 

x'  — 3axi/ +  t/^*  =  0.  r" 

Sr"  —  Say  =  o^  '''  V  =^  ~^  •'•  ^  ~  2«'ar'  =  0 

.-.  X  =  0  orx  =  a  3/ 2  ••.  (52) 

(Py  ^cPu  ^         du  X*  20,'  ^ 


=  -  or 
a 


3/2  —  1 

.•.  when  X  =  0  ....  1/  =  0,  a  minimum. 

X  =  0^2  ....  1/  =  a  ^4,  a  maximum. 

10.  To  divide  a  given  number  a,  into  two  parts,  such  that  the 
product  of  the  mth  power  of  the  one  and  the  nth  power  of  the  other 
shall  be  the  greatest  possible. 

Let  X  be  one  part,  then  a  —  x  is  the  other,  and 

t/  =  x"  (a  —  x)"  =  maximum, 

.'.  -p  =  mx"~'  (a  —  x)"  —  nx"  (a  —  x)"-' 

=  x^'  {a  —  x)"-'  \ma  —  {m  -\-  n)  x\  =  0, 
.'.  X  =  0,  or  o  —  X  =  0,  or 


THE    DIFFERENTIAIi    CALCULUS.  75 

ma  —  {r.i  +  n)  a;  =  0, 
which  give 

ma 

X  =  0,x  =  a,x  = ; . 

m  +  n 

The  first  and  second  of  these  values  are  inadmissible,  because  the 
number  is  not  divided  when  x  =  0  or  when  x  =  a. 
Substituting  the  third  value  in 

— i-  =  x""'  (o  —  x)**"*  \{ma  —  {m  -]-  n)  xy  —  m{a — xY  —  nx-| 
oar 

we  have 

^^^  ^  -  W"-^  [«  -  ^T-'  ^»  [«  -  ^T  +  ^^\ 

which  is  negative  because  each  factor  is  positive,  hence  the  two  re- 
quired parts  are 

ma  ,      na      ,    .  ,      , 

and  — -j- —  bemg  to  each  other  as  m  to  n. 


m  +  »        7n  -\-  n 
Cor.  If  TO  =  n  the  parts  must  be  equal. 
An  easier  solution  to  this  problem  may  be  obtained  as  follows : 

Put  —  =  p  and  determine  x  so  that  we  may  have 
n 

M  =  arP  (a  —  x)  =  A  maximum, 

'  du  ,  ,  ^ 

=  xP-'  \pa—(p  -jr  \)cr\  ^  0, 

pa 

.-.  X  =  0  —  or  pa  —  (p  +  1)  X  =  0  .-.  X  =  — ij — . 

P  +  1 

This  last  value  substituted  in 

_  =  r^-^  ^pa  —{p+i)xl—{p-\-l)  X'-' 

causes  the  first  term  to  vanish ;  the  result  is  therefore  negative,  so 

pa  ma  ,     ^  •  ,        ^ 

that  X  =  ~ = ; —  corresponds  to  a  maximum  value  of «, 

p  +  I       m  +  n 

and  therefore  (53)  to  a  maximum  value  of  m"  —  x"  {a  —  x)". 

Another  easy  mode  of  solution  is  had  by  using  logarithms,  for  it  is 


70 


THE  mrFERBMTlAL  CALCULUS. 


obvious  that  sinco  the  logarithm  of  any  number  increases  with  the 
number,  when  this  number  is  the  greatest  possible,  its  logarithm  will 
be  so  also. 

.'.  in  log.  jj  +  n  log.  (a  —  x)  ~  max. 

du Ml  n 

da        X        a  —  X  \       >      j 


tn  +  « 
as  before. 

The  expression  for  the  second  differential  coefficient  is  —  (tn  + 
n)  showing  that  the  foregoing  value  of  x  renders  the  logarithmic  ex- 
pression a  maximum. 

1 1 .  To  divide  a  number  o,  into  so  many  equal  parts,  that  their 
continued  product  may  be  the  greatest  possible. 

It  is  obvious  from  the  corollary  to  the  last  example,  that  the  parts 
must  be  equal,  for  the  product  of  any  two  unequal  parts  of  a  number, 
is  less  than  that  of  equal  parts. 

Let  X  be  the  number  of  factors,,  then,^ 


0- 

=  o,  maximum, 

...  log. 

0-  = 

=  a:  log.  {-) 

=  a  maximum,. 

••• 

log.-_l 

=  0 

a 

X 

=  log. 

-'1  =  e.-. 

a 

X  —-* 
e 

hence  the  proposed  number  must  be  divided  by  the  number  e  = 
2-718281828. 

12.  To  determine  those  conjugate  diameters  of  an  ellipse  which 
include  the  greatest  angle. 

Call  the  principal  semi-diameters  of  the  ellipse  a,  b,  the  sought 
semi-conjugates  x,  x'  and  the  sine  of  the  angle  they  include  y.  Then 
{Anal.  Geom.) 

*  There  is  obviously  no  necessity  to  recur  to  the  second  differential  coefficient 
to  ascertain  whether  this  value  render  the  function  a  maximum  or  a  minimum, 
since  it  js  plain  that  there  is  no  minimum  unless  each  of  the  parts  may  bo  0. 


THB  DIFFERENTIAL  CALCULUS.  77 


T^  +  «"  =  a^  +  6=  .-.x  =  ^^a?  ■{■  y  —  ir 

ah 
xxy  =^  ao  .'.  y  =^  — — 

''  ^       xx' 

,  ab 

.  •.  t/  = : =  max. 

^       xV  a'  +  b^—sr' 

Omitting  the  constant  06,  inverting  the  function  (ex.  6.)  and  squar- 
ing, we  have 

u  =  oV  +  b^ar^  —  x*  =  max. 

du 
...  —  =  2o='a;  +  2b^x  —  4x'  =  0, 
ax 

fjfi   4-    A3  3      1         '2 

2  2 

The  first  of  these  values  is  inadmissible,  from  the  second  we  find 
that 

hence  the  conjugates  are  equal.     For  the  second  differential  coefii- 
cient  we  have 

-r^  =  2a=^  +  2b-  —  Ux" 


This  being  negative,  shows  that  x  =  V corresponds  to  a 

maximum  value  of  m,  or  to  a  minimum  value  of  j/,  so  that  tho  conju- 
gates here  determined,  include  an  angle  whose  sine  is  the  least  pos- 
sible; and  this  happens  when  the  angle  itself  is  the  greatest  possible 
(being  obtuse),  as  well  as  when  it  is  the  least  possible. 

13.  To  divide  an  angle  &  into  two  parts,  such  that  the  product 
of  the  nth  power  of  the  sine  of  one  part  of  the  >«th  power  of  the  sine 
of  the  other  part  may  be  the  greatest  possible. 
Let  X  be  one  part,  then  &  —  a;  is  the  other,  and 

sin. "a; .  sin.*"  {&  —  x)  =  maximum, 
.  ••  »  log.  sin.  X  +  m  log.  sin.  (^  —  x)  =  maximum, 
n  cos.  X       in  cos.  {&  —  x) 
sin.  X  sin.  (d  —  x) 


78  THK  DIFFERENTIAL  CALCL'LUS. 

.'.  ntan.  {6  —  x)  —  m  Ian.  x, 

.'.  n  :  HI  :  :  tan.  x  :  tan.  {6  —  x), 

.'.  n  -}"  ni  :  "  —  "*  : :  tan.  x  +  tan.  {6  — x)  :  tan.  x —  tan.  (6  — x), 

: :  sin.  d  :  sin.  {'2x  —  6),* 

.-.  sin.  (2x —  6)  — ■ sin.  6, 

n  +  m 

which  determines  x. 

14.  Given  the  hypothenuse  of  a  right-angled  triangle  to  deter- 
mine the  other  sides,  when  the  surface  is  the  greatest  possible. 
Call  the  hypothenuse  a,  and  one  of  the  sides  x,  then  the  other  will 

be  Va'  —  ^  and  the  area  of  the  triangle  will  be 


^  %/  a^  —  r*  =  maximum. 
.•.  u  =  a^x*  —  X*  =  maximum. 

t^"  «     o  .      1  ^  r.  « 

.•,-r-  =  2a-x  —  4:X   =  0  .'.  a:  =  0  or  X  =  — r-. 
ax  V2 

Substituting  the  second  value  in 

=z  2aF—  12x2 

the  result  being  negative,  shows  that  the  above  value  of  x  corresponds 

a 
to  a  maximum.     Therefore  the  required  sides  are  each  —r~. 

V  2 

1 5.  To  determine  the  maxima  and  minima  values  of  the  function 

1/  =  x^  —  ISx^  +  96x  —  20. 
when  X  =  4t  .  .  .  .  y  =  356  a  maximum. 
X  —  8... .7/  =  128  a  minimum. 

16.  To  determine  a  number  x,  such  that  the  ath  root  may  be 
the  greatest  possible. 

Ans.  x  —  R—  2-71828  .... 

17.  What  fraction  is  that  which  exceeds  its  nith  power  by  the 
greatest  possible  number  ? 

m— 1   2 

Ans.  \f — . 
m 


*  Dr.  Giegory'i  Trigonometry,  p.  47,  Equation  (S). 


THB  DIFFERENTIAL  CALCULUS.  19 


18.  Given  the  equation 

y^  —  2mxy 

+ 

ar^  =  a^, 

to  determine  the 

maxima  and  minima  values  of  y. 

When  X  = 

ma 

y 

a 

a  maximum, 

v/  1— TO- 

VI— ni"' 

X  —- 

—  ma 

y 

—  a 

,  a  minimum. 

V  1  _  ,rt3  •'         ^  i  —  m^ 

19.  Given  the  position  of  a  point  between  the  sides  of  a  given 
angle  to  draw  through  it  a  line  so  that  the  triangle  formed  may  be  the 
least  possible. 

Ans.  The  line  must  be  bisected  by  the  point. 

20.  The  equation  of  a  certain  curve  is  a^j  =  ax^  —  x^  required 
its  greatest  and  least  ordinate  s. 

When  x  =  |a  .  .  .  .  J/  =  maximum, 
at  =  0    .  .  .  .  t/  =  minimum. 

21.  To  divide  a  given  angle  d  less  than  90°  into  two  parts,  x  and 

t  —  X,  such  that  tan."  x  .  tan."*  (^  —  x)  maybe  the  greatest  possible. 

n  —  m 
tan.  (2x  —  &)  = i tan.  6. 

22.  To  determine  the  greatest  parabola  that  can  be  formed  by 
cutting  a  given  right  cone.* 

SCHOLIUM. 

(65.)  It  will  be  proper,  before  terminating  the  present  chapter, 
to  apprize  the  student  that  in  the  application  of  the  theory  of  maxima 
and  minima  to  geometrical  inquiries,  care  must  be  taken  that  we  do 
not  adopt  results  inconsistent  with  the  geometrical  restrictions  of  the 
problem.  We  know,  indeed,  from  the  first  principles  of  Analytical 
Geometry,  that  when  the  geometrical  conditions  of  a  problem  are 
translated  into  an  algebraical  formula,  that  formula  is  not  necessarily 
restricted  to  those  conditions,  but,  in  addition  to  all  the  possible  solu- 
tions of  the  problem,  may  also  furnish  others  that  belong  merely  to 
the  analytical  expression,  and  have  no  geometrical  signification.!  If, 

*  It  will  be  shown  hereafter  that  a  parabola  is  equal  to  J  of  a  rectangle  of  the 
same  base  and  altitude, 

\  See  the  Analytical  Geometry. 


80  THE  DIFFERBNTIAL  CALCULUS. 

therefore,  among  these  latter  solutions  there  be  any  belonging  to 
maxima  or  minima,  they  are  inadmissible  in  the  application  of  this 
theory  to  Geometry.  The  following  example  is  given  by  Simpson, 
at  art  47  of  his  Treatise  on  Fluxions,  to  illustrate  this. 

-S^F         From  the  extremity  C  of  the  minor  axis  of  an  €lhpse 
,y  |\     to  draw  the  longest  line  to  the  curve.     Suppose  F  to  be 
jj    the  point  to  which  the  line  must  be  drawn,  and  call  the 
'     abscissa  CE,t  then  the  geometrical  restrictions  of  this 
variable  are  such  that  its  values  must  always  lie  between 
the  limits  x  =  0  and  x  =  26,  a  and  b  denoting  the  semi- 
axes. 

By  the  equation  of  the  corve. 

EF'  =  f  =  ^i2bx-:c') 

a" 
...  CF^  —  w  —  ^  +  IT  (2^^  —  xF)  =  maximum. 

du       „  ,      ,    "^        d^     , 


d'—b^ 


and  since  -r-^  =  2  (1  —  y^)  it  follows  that  the  foregoing  expres- 
sion for  x  renders  u  a  maximum  for  all  values  of  6  less  than  a,  and  a 
minimum  for  all  values  of  b  greater  than  a.     Hence  if  the  relation 

n-h 

between  a  and  b  be  such  that  — r^  may  exceed  26,  the  analytical 

expression  for  CF  will  admit  of  a  maximum  value,  although  such 

value,  not  coming  within  the  geometrical  restrictions  of  the  problem, 

a^b 

is  inadmissible.    If  the  relation  between  a  and  6  be  such  that  — j-„ 

a-'  —  6^ 

=  26,  that  is,  if  a^  =  26^  the  solution  will  be  valid,  and  in  the  ellipse 

whose  axes  are  thus  related  CD  will  be  the  longest  line  that  can  be 

dravm  from  C,  agreeably  to  the  analytical  determination,  and  the 

solution  will  always  be  vahd  if  the  axes  of  the  ellipse  arc  related  so 

a^b 

that p;  is  not  greater  than  26,  which  leads  to  the  condition  26^^ 

a" —  6" 

not  greater  than  a^. 


THE    DIFFERENTIAL    CALCULUS.  81 


CHAPTER   VII. 

ON  THE  DIFFERENTIATION  AND  DEVELOPMENT 

OF  FUNCTIONS  OF  TWO  INDEPENDENT 

VARIABLES. 

Differentiation  of  functions  oftivo  independent  variables. 

(56.)  Let  z  =  F  {x,  y)  be  a  function  of  two  independent  varia- 
bles ;  then  since  in  consequence  of  this  independence,  however  either 
be  supposed  to  vary,  the  other  will  remain  unchanged  :  the  function 
ought  to  furninsh  two  differential  coefficients  ;  the  one  arising  from 
ascribing  a  variation  to  x  and  the  other  from  ascribing  a  variation  to 
y,  y  entering  the  first  coefficient  as  if  it  were  a  constant,  and  x  enter- 
ing the  second  as  if  it  were  a  constant.     The  differential  coeffi- 

dz 
cient  arising  from  the  variation  of  a:  is  expressed  thus,  — ;  and  that 

arising  from  the  the  variation  of  y  thus,  — ;  and  these  are  called  the 

partial  differential  coefficients,  being  analogous  to  those  bearing  the 
same  name  considered  in  chapter  IV.  We  have  seen,  in  functions 
of  a  single  variable,  that  if  that  variable  take  an  increment,  and  the 
function  be  developed,  what  we  have  called  the  differential  coefficient 
will  be  the  coefficient  of  the  first  power  of  the  increment  in  that  de- 
velopment ;  so  here,  as  will  be  shortly  shown,  the  partial  differential 
coefficients  are  no  other  than  the  coefficients  of  the  first  power  of 
the  increments  in  the  development  of  the  function  from  which  they 

dz 
are  derived.     As  to  the  partial  differentials  they  are  obviously —  dx 

and  -J-  dy  and  hence  we  call  -r-  dx  -^  -r-  dii  the  total  differential 
ay  dx  dy  "^ 

of  the  function,  that  is, 

dz  =  -r-  dx  +  -r-  dy, 
dx  dy 

and  we  immediately  see  that  this  form  becomes  the  same  as  that 

11 


82  THE  DIFFERENTIAL  CALCULUS. 

given  in  chapter  IV.  for  the  differential  of  F  {x,  q)  as  soon  as  we 
suppose  7/  to  be  a  function  of  x,  for  we  then  have 

dz . dz       dz     dy 

^ux        dx       dy     dx^ 
as  indeed  we  ought. 

In  a  similar  manner,  if  the  function  consist  of  a  greater  number  of 
independent  variables  as  «  =  F  (x,  y,  z,  &c.)  we  should  necessarily 
have  as  many  independent  differentials,  of  which  the  aggregate 
would  be  the  total  differential  of  the  function,  that  is 

du  =^  -r-  dx  +  -r-  dy  -\-  ^-  dz  -\r  &c, 
dx  dy  dz 

Hence,  whether  the  variables  are  dependent  or  independent,  we 
infer,  generally,  that 

The  total  differential  of  any  function  is  the  sum  of  the  several 
partial  differentials  arising  from  differentiating  the  function  relatively 
to  each  variable  in  succession,  as  if  all  the  others  were  constants. 

We  shall  add  but  few  examples  in  functions  of  independent  varia- 
bles, seeing  that  the  process  is  exactly  the  same  as  for  functions  of 
dependent  variables. 

d  {x  -^r  ]})  =  dx  -^r  dy 
d  .  xy  ^  ydx  +  xdy 
1  X  _  ydx  —  xdy 

y  ¥ 

ay  as^dy  —  ayxdx 


Vxr'  +  y''  (^  +  2/')* 

,  .        ,  X  ydx  —  xdy 

d  tan.~'  -  = 

y  f  -V  3? 


d 


y        _  yd^  —  ^y^dy  —  xdy 


3j/2— X  {^Mf  —  xf 

d.  a'  b^  c'  =  a'b^  c''{dx  log.  a  ■\- dy  log.  h-{-dz  log.  c) 
d  log.  tan.  -  =      ^^-^  —  ^^y      =  2  {ydx  —  xdy) 

y      o  ■    X       X         „  .    2x 

y   sm.  -  cos.  -  w^  sm.  — 

y       y        '^        y 

dy'  =  f  log.  ydx  +  I/*-'  xdy. 
(57.)  If  the  function  that  x,  2/  is  of  2  is  given  implicitly,  that  is  by 
the  equation 


then 


but  (39), 


THE  DIFFERENTIAL  CALCULUS.  88 

«  =  F  {x,  y,  z)  =  0, 


■du^^        „       ,  .du. 


^dx^ 

du       du 
dx       dz 

dx 

du       du 
dy       dz  ' 

■?  =  « 

dy 

,  Au    ,    du     dz^    ,      ,    ,du    ,    du     dz         _ 

^dx       dz     dx'  dy       dz     dy'    -^ 

Thus :  let  Ax"  +  Bi/='  +  Cc^  _  i  =  o, 

.-.  du  =  {Ax  +  Cz^)  dx  +  (By  +  Cz^)  dy  =  0. 

(58.)  If «  =  Fz,  z  being  a  function  of  x  and  y,  the  two  differen- 
tial coefficients  are  (33) 

du  _  du    dz  du  _  du    dz 

dx       dz  '  dx*  dy       dz  '  dy 
and  the  total  differential  is,  therefore, 

*  The  brackets  arc  employed  here  for  the  same  purpose  as  at  (37),  viz.  to  im- 
ply the  total  differential  coefficient  derived/ram  u,  considered  as  a  function  of  a  single 
variable.  This  form  it  will  be  necessary  to  adopt  whenever  m  contains,  besides  x, 
other  variables  that  are  functions  of  x,  provided  we  wish  to  express  the  total  coeffi- 
cient with  respect  to  a:.  No  ambiguity  can  arise  from  our  calling  these  same  coef- 
ficients partial  in  one  sense,  and  total  in  another.  They  are  partial  coefficients  in 
relation  to  the  whole  variation  of  i«,  but  they  are  total  coefficients  as  far  as  that 
variable  is  concerned  whose  differential  forms  the  denominator;  and  it  may  be  re- 
marked here,  once  for  all,  that  when  we  enclose  a  differential  coefficient  in  brack- 
ets, we  mean  the  tot(d  differential  coefficient  to  be  understood,  arising  from  consi- 
dering the  function,  whose  differential  is  the  numerator,  as  simply  a  function  of  th« 
variables  whose  diflerentials  form  the  denominator. 


84  THE  DIFFERENTIAL  CALCULDS. 

-        du    dz  -     ,   du    dz 

dU   =  -r-  .   -r-  dX  -{-  -r-  .   -f-  AV. 

dz     dx  dz     drj 

Now  it  is  worthy  of  notice,  that  the  ratio  of  the  tieo  partial  differ- 
ential coefficients  is  independent  of  F,  so  that  this  may  be  any  func- 
tion whatever.     Thus 

du    ,    du du     dz    _    du     dz  dz   .    dz 

dx  '  dy  dz'  dx  '  dz'  dy  dx  '  dy 
which  is  an  important  property,  since  it  enables  us  to  eliminate  any 
arbitrary  function  F  of  a  determinate  function  y'(,r,  ij)  of  two  variables. 
We  shall  often  have  occasion  to  employ  it  in  discussing  the  theory  of 
curve  surfaces.  By  means  of  this  property  too  we  may  readily  as- 
certain whether  an  expression  containing  two  variables  is  a  function 
of  any  proposed  combination  of  those  variables.  For,  calling  this 
combination  z  and  the  function  «,  we  shall  merely  have  to  ascertain 
whether  or  not  the  above  condition  exists,  or,  which  is  the  same  thing, 
whether  or  not  the  condition    . 

du    dz       du    dz  _ 
dx  '  dy       dy'  dx 
exists.     For  instance,  suppose  we  wished  to  know  whether  m  =  a?* 
-f-  2x^f  +  ?/*  is  a  function  o^  z  ^=  oc^  ■{•  y^. 
Here 

^«        .  ■,   I    .     »  ^^'*        .  o     I    .  -^    o,z        ^     dz 

-  =  4.^  4-  4:ry^-  =  ^x^y  +  Ay^,-  =  2y,-=  2x; 

du    dz       du     dz        ,,■,,.     n\  ^  ,  ^  n     ,    .  t^  ^ 

dx    dy       dy     dx        ^  j  ^    j        v      :;  n  j 

consequently,  since  the  proposed  condition  exists,  we  infer  that  u  is 
a  function  of  x. 

We  shall  now  proceed  to  apply  Taylor's  theorem  to  functions  of 
two  independent  variables. 

Development  of  Functions  of  two  Independent  Variables. 

(59.)  In  the  function  z  =  F  (x,  y)  suppose  x  takes  the  increment 
h,  the  function  will  become  F  (a;  +  h,  y),  y  remaining  unchanged, 
since  it  is  independent  of  x,  then,  by  Taylor's  theorem, 

dz  d-z        h^  d^z  h^ 

&c.  .  .  .  (1). 


THE  DIFFERENTIAL  CALCULUS.  85 

But  if  y  also  take  an  increment  k,  then  z  will  become 

dij  dif      1*2        dif      1  •  2  •  d 

dz 
so  that  in  the  expression  (1)  we  must  for  -j-  substitute 

dz  d^z  d?z 

dz  djj  If      F  _LS.       ^'       ,   o 

dx  "^       c/x      ^'+      rfo;      •1-2'^         dx      •  1.2-3  +  ^^' 

dz  d^z  d^z 

d^z        "^   '  dy         "*       dy'     Jf_         _J_f_        ^-^ 

d^  dFz  cPz 

d^z  '  dy  '  dw^        k^  '  dy^  P 

1 ±  h  J ±—     4- i_ L    fop 

dx"  ^      dar"    '^^      dr*       •1-2^       dx"       •l-2-3^*^*^' 

and  so  on.     Before,  however,  we  actually  make  these  substitutions, 
we  shall,  for  abridgment,  write 

dz  drz 

d'z  ^  '   dji  _^z__         ^'  df  di^z 

dydx  dx     '  dy^  dx  dx  S  7  ^yidxP 

d'z 

for  •' 


dxP 

this  last  expression  implying  that  after  having  determined  the  qth 
differential  coefficient  of  the  function  z  relatively  to  the  variable  i/, 
the  j9th  differential  coefficient  of  this  is  taken  relatively  to  the  other 
variable  x.  Hence,  the  result  of  the  proposed  substitutions  in  (1) 
will  be 

.     ¥{x-\-h,y^h)  = 


86 


THE  DIFFERENTIAL  CALCULUS. 


ax 


dz 
dy 


k 


+ 


d'^z       h^ 


dx^  *  1  •  2 

dydx 
dh        ¥ 


dtf   '1-2 


c/y  dx^ 
d'z 

d^f  dx 

i.^z 


df 


1  •2-3 

hh~ 

1  •  2 
¥h 

1  -2 
1-2-3 


+  &C. 


The  general  term  of  the  development  being 


di/*  d.i''  '  (1  •  2  .  .  .  5)(1  -2  .  .  .pY 
If  in  the  proposed  function  z  =  'F  {x,  y)  we  had  supposed  y  to  vary 
first,  then,  instead  of  (1),  whe  should  have  had 

^■2       .   d'z  P 


+  &c.  .  .   .   (2). 
But,  if  a:  take  the  increment  h,  z  will  become 


dz  ,    ,    d'z        h^ 

z  +  —  h  •+■  . 

dx     ^  d3^     1-2 


+ 


d'z 

df'  l-2'3 


+  &C. 


dx'      1  •  2  •  3 
and,  therefore,  we  must  substitute,  agreeably  to  the  foregoing  nota- 


tion for 


dz 
dy 


for 


dz    .      d^z   , 
dy       dxdy 
di'z 


dh 


+ 


d'z 


da^  dy  *  1  •  2         dx'^  dy      1  •  2  •  3 


+  &c. 


df 

dPz        _fz__  ^^  d'z 

df        dx  dy^  dx^  df 


h^  d'z 


K' 


1  •  2       dx^dy'' '  1  •  2  •  3 


+  &C. 


for 


df 
d'z 


d^z      ,     ,       d^z 
h  + 


Iv" 


+ 


d'z 


df    '   dx  dy"^ '"    '    dir  dy'  *  1  •  2       dx'  df  '  1  •  2  •  3 
and  so  on ;  so  that  the  development  would  be 
F  {x -\- h,  y  -\-  k)  = 


+  &c. 


THE  DIFFERENTIAL  CALCULUS. 


87 


,dz  , 

dx 
dz 

dx 


-t- 


d"-z 

Iv" 

dor 

1  .  2 

d'z 

dxdij 
d'z 

hk 

df 


1  .  2 


+ 


d'z 

h^ 

dx'      ' 
d'z 

1  •2-3 

]rk 

dx^  dy  ' 
d'z 

dx  dif  ' 
d'z 

1  •  2 

1  •  2 

df 


1  •  2  -3 


+  &C. 


hP  .  h'i 


the  general  term  being 

d'-^z 

~dxP  df    '  {1  -2.  .  .p)    (1  -2  ..  .  q)' 
As  this  development  must  be  identical  with  that  exhibited  above, 
we  have,  by  equating  the  like  powers  of  h  and  A;, 
d'z   _     d^z       d^z     _    dPz 
dijdx       dxdij   dy  dx^       dx^  dy 
and  generally 

d^-^z  _  dP-^z 
dif  dxP  ~  dxP  dy'' ' 
we  conclude,  therefore,  thatif  we  first  determine  the  gth  differential 
coefficient  relatively  to  the  variable  y,  and  then  the  pih.  differential 
coefficient  of  this  relatively  to  the  variable  x,  the  final  result  will  be 
the  same  as  if  we  first  determine  the  jpth  differential  coefiicient  rela- 
tively to  Xy  and  then  the  qth  differential  coefficient  of  this  relatively 
to  y  ;  so  that  the  result  is  the  same  in  whichever  order  the  differen-. 
tiations  are  performed. 

(60.)  We  see  from  the  foregoing  development,  that  the  partial 
differential  coefficients  of  the  first  order  are  the  coefficients  of  h  and 
k,  the  first  power  of  the  increments,  so  that  the  term  containing  these 
first  powers  is  in  this  respect  analogous  to  that  containing  the  first 
power  of  the  increment  in  the  development  of  functions  of  a  single 
variable,  and,  by  a  very  slight  transformation,  it  will  be  seen  that  the 
same  analogy  extends  throughout  a  1  the  terms  of  the  two  develop- 
ments. For  the  development  just  given  may  be  put  under  the  form 
Fix  +  h,y  +  k)  = 
z 


+(^^  +  1*) 


^dx 


88  THE  DIFFERENTIAL  CALCULUS. 

+    _J_  f^  7,3    .    2    '^'^      hk  4-         ^^      k") 

,        1      ,d^2   ,„  ,  „    d^z      ,,,    ,  „    rf^2      ,  ,,     ,     d^z   J.. 
2-3  VjH  du^  dij  dx  dy^  dx'      ' 

+  &c. 

where  the  partial  differential  coefficients  in  each  term  are  identical 
with  those  which  appear  in  the  differential  of  the  preceding  term,  as 
the  actual  differentiation  shows,  thus  : 

dz  —  -^dx-\-~dy.  .  .   .   (1), 
dx  dij 

the  coefficients  y-,  -^,  being  functions  of  x  and  y,  we  have 

dz  _    d'z  d'z 

dx         dx^  dxdy 

dz  _  d^z  d'^z 

dy       dydx  dif 
and,  consequently, 

In  like  manner,  these  coefficients  being  functions  of  a:  and  y,  we 

have 

d'z    _     d'z  d'z 

dx^  dx'  dj^  dy 

d'z    d'z  d'z 

dxdy        dx^  dy         dx  dy^ 


,        d'z  d'z        ,       d'z         ■ 

d  '  —r^r  =    ,  .,  ,      +  — r-, —  dy 

dy^  dy  dx  dif 


so  that 


'^"y-  ■  ■  '■  ■  (^'' 

and  so  on ;  the  numeral  coefficients  agreeing  with  those  in  the  cor- 
responding powers  of  the  expanded  binomial. 

(61.)  Having  now  applied  Taylor's  theorem  to  functions  of  two 


THE  DIFFERENTIAL  CALCULUS.  80 

Variables,  we  may  equally  extend  Maclaurin's  Theorem.  For,  if  in 
the  foregoing  development,  we  suppose  x  and  y  each  =  0,  the  de- 
velopment will  become  that  of  the  function  F  {h,  k)  according  to  the 
powers  of  h  and  k  ;  or,  substituting  x  and  y  for  the  symbols  h  and  k, 
since  these  are  indeterminate,  we  have 

The  principles  by  which  we  have  thus  extended  the  theorems  of 
Taylor  and  Maclaurin  are  sufficient  to  enable  us  to  extend  these 
theorems  still  further,  even  to  the  development  of  functions  of  any 
number  of  variables  whatever,  but  this  is  unnecessary.  It  maybe 
remarked,  however,  that  if  we  wish  to  develop  a  function  of  several 
variables  according  to  the  powers  of  one  of  them,  it  may  be  done 
independently  of  any  thing  taught  in  this  chapter ;  for,  if  all  the  varia- 
bles but  this  one  were  constants,  the  development  would  agree  with 
that  already  established  for  functions  of  a  single  variable,  and,  as 
these  constants  may  take  any  value  whatever,  they  may  obviously  be 
replaced  by  so  many  independent  variables.  We  shall  give  one 
instance  of  this  extension  of  Maclaurin's  theorem  to  a  function  of  two 
independent  variables,  choosing  a  form  of  extensive  application  and 
of  which  the  development  is  known  by  the  name  of 

Lagrange's  Theorem. 

(62.)  The  function  which  we  here  propose  to  develop  according 
to  the  power  of  x,  is 

M  =  Fz,  in  which  z  =  y  -\-  xfz, 
z  being  ob^ously  a  function  of  the  independent  variables  x  and  y. 
We  shall  first  develop  z  =  y  -{-  xfz  according  to  the  powers  of  x  : 
this  development  is  by  Maclaurin's  theorem 

dz  d'z       x^      ,     d^z  T       x^ 

'  =  W  +  ts^  -  +  [5?]  —2  +  fe-]  FFTa  +  «"=• 

and  if  we  denote  according.to  the  notation  of  Lagrange  the  successive 
differential  coefficients  of/z,  relatively  to  x  hyf'z,f"z,J"'Zf  &c.  we 
shall  have 

12 


90  THE  DIFTERENTIAL  CALCULUS, 

dz 

&c.  &c. 

Consequently,  when  a;  =  0, 

W    =y 
^'d^^~^-d^J'J      df 

^da^^-^^^jy^    ^  ^  •  ~di, d^  ^ 

&c.  &c. 

Hence 

n       J        u  -yjj    ^-y      ^y        1  .  2  ^      dy'       1  •  2  •  3 

+  &C (1). 

Now,  instead  of  this  development,  we  should  obviously  have 
obtained  that  of  Fz  =  F  (]/  +  xfz),  if  in  place  of  ^  and  its  differen- 
tial coefficients  we  had  employed  Fz  and  its  differential  coefficients. 
We  should  then  have  had 

u  =  F  {y  +  xfz)      .     .     therefore  .     .     [  m  ]  =  Fy 
du  _du     dz  du     _  dFy 

dx~ds'dx *■  dx^~  dy  ^y 

d'u_  d^u   dz  du     dPz  r^i —^Fl-f  f  \a  4. 

d?~'d^^dx^   "^   dz'  dx" '-(/a^-'        dy"  ^^^  "^ 

dFy 

dFy    d.{fyf    ^  ^•'dj^fy^' 
dy   '      dy  dy 

&c.  &c. 

HeQC« 


THE  DIFFERENTIAL  CALCULUS.  91" 

Tz  =  F  (y  +  xfz)  =  Fw  +  — -^  fy  •-  + ; . + 

and  this  is  Lagrange's  Theorem.* 

From  this  remarkable  expression,  which  includes  that  marked  (1), 
other  forms  may  be  readily  deduced  as  particular  cases.  Of  these 
the  two  following  are  the  most  important. 

Put  a?  =  1,  then  the  formula  (1)  becomes 

z  =  y-\-fz  =  y-{-fy  H hLLL. . L  — \dAL  . 4- 

H    i  J  a       JiJ-r      ^^         1  .  2  ^      dy''         1  •  2  •  3 

&c (3), 

and  the  formula  (2), 

d.Fy             "*•     dy    ^^y^  1 

„  =  F(y+/.)=F,  +  -^//,+  ^_._  + 

^.^iJyf       , 

la-^ — •  TT^  +  &^ (4). 

(63.)  We  shall  terminate  the  present  section  with  one  or  tAvo  ex- 
amples of  the  application  of  these  formulas,  referring  the  student  for 
more  ample  details  on  this  subject  to  Lagrange's  Resohilion  des 
Equations  J^Tumeriques,  note  xi. ;  and  Jephson's  Fluxional  Calculus, 
vol.  i. 

EXAMFLES. 

1.  Given  ^  —  52  +  r  =  0,  to  develop  z  according  to  the  pow- 
ers of  r. 

Since  here2  =  --] ,wehavey  =  -,  fz  =  -  2^ .:  fy  =^  -  (-Y 

q        q  -^        q'J         q  J^       q^  q> 

*  For  another  and  very  complete  demonstration  of  this  theorem  see  note  (Bl 
at  the  end. 


M  THE    DIFFERENTIAL    CALCULUS. 

.difyy_  1    d.t/_  6   3  dFjfyY  _  1    ^Y  _  9  /^l/'  _ 
di/  9^  '   dy         q^^ '     df  q^  '  dy^       f  '  dy 

Hence,  by  the  formula  (3),  we  have,  by  putting  for  y  its  value  - 

=  !:4.i  rl-i-—!—  '"  I-  ^  •  ^     li  +  fe 

q  q^        1   •  2^'       1  •  2  •  9" 

2.  Given  the  radius  vector  of  an  ellipse,  viz.  {Anal.  Geom.) 

1  —  e^ 

r  =  a  . 

1  +  e  COS.  u 

to  develop  r",  according  to  the  powers  of  cos.  w. 

Since  r  =  o  (1  —  c-)  —  e  cos.  w  .  r,  we  have,  by  putting  y  for 
0(1  —  e^)  and  ar  for  —  e  cos.  cj, 

Fr  =  F  (t/  +  x/r)  =  F  (y  +  X  •  r)  =  (y  +  X  •  r)". 
Hence,  by  the  formula  (2) 

dy'  X  dy  ^         x^ 

'"  =  J/"+%-  2'-  1  +         dy        -1^  + 

dy  ^  x' 

=  y-  +  ny .  I  +  n  (n  +  1)  y"  .  ^p-_  + 
n{n-{-  l)(n  +  2)t/".^-^+  &c. 


=  a-  ( 1  -' e^)"  ( 1  -  "^^^^^  +  ^  ^^  "^  ^^ 


C  COS."  W  — 


n(n+l)(n  +  2)    ,        , 

fTY-l '^  <^os-  <^  +  &c. 

8.  It  is  required  to  revert  the  series 

a  +  I3z  +  ys?  ■{■  dz' +  &,c.  =  0, 


THE    DIFFERENTIAL   CALCULUS.  93 

that  is,  to  express  the  valufe  of  z  in  terms  of  the  coefficients.     Here 

2  =  — 1--|- (7  +  ^^  +  &c.)  =  2/ +/^ 
therefore,  by  the  formula  (3), 

^  d.^,(r  +  5,/  +  &c.)^     ^ 

d^  "I,  (r  +  ^2/ +  &c.)^       J 

1^  (7  +  %  +  &c.)  (5  +  2sy  +  &c.) 
-^  6  |j  (7  +  5y  +  &c.)='  +  &c. 

+  &C. 

where  t/  = ^  .  consequently 

4.  Given  1  —  z  +  az  =  0  to  develop  log.  z,  according  to  the 
powers  of  a. 

log.  2  =  a  +  1  o^  +  1  a^  +  i  a*  +  &c.* 

*  This  we  know  from  other  principles ;  for,  since  the  proposed  expression  re- 
dncestoz  =  — — -  .•.  log.  z  =  —  log.  (1  —  a)  and  this,  in  the  hyperbolic  system, 
is  equal  to  the  above  series.     (See  the  Essay  on  Logarithms,  p.  3.) 


94  THE  DIFFERENTIAL  CALCULUS. 


CHAPTER  VIII. 

ON  THE  MAXIMA  AND  MINIMA  VALUES  OF  FUNC- 
TIONS OF  TWO  VARIABLES,  AND  ON  CHANGING 
THE  INDEPENDENT  VARIABLE. 

(64.)  It  remains  to  complete  the  theory  of  maxima  and  minima 
by  applying  the  principles  established  in  Chapter  VI.  to  functions  of 
two  independent  variables. 

The  same  character  belongs  to  a  maximum,  or  minimum  function 
of  two  variables  that  belongs  to  a  maximum  or  minimum  function  of 
one  variable,  that  is,  the  maximum  value  exceeds  the  contiguous  va- 
lues of  the  function,  and  the  minimum  value  falls  short  of  them. 

Hence,  if 

2  =  F  {x,y) 

be  any  function  of  two  variables,  wliich  becomes  a  maximum  for  cer- 
tain particular  values  of  them,  then  h  and  Jc  being  finite  increments, 
however  small  the  condition  is  that,  between  such  finite  values  and 
0,  we  must  always  have 

Flx,9j-]>F[x±  h,y±  kl 
and,  consequently,  (60), 

(±£*±?/)+*(^*'±^^"+^*=)+^-<''-* 

If,  therefore,  of  the  small  values  which  we  suppose  h  and  k  to  take, 
h  be  the  smallest,  a  part  of  k  maybe  taken  so  small  as  to  be  less  than 
h,  or,  which  is  the  same  thing,  equal  to  one  of  the  values  of  h  between 
the  proposed  value  and  0,  so  that  we  have  h'  —  k' ;  therefore,  the 
above  condition  is 

'-        dx       dx  '    dar^  dx  dy       dy'^ 

This  condition  being  similar  to  (1)  art.  (49),  we  infer,  by  the  same 
reasoning,  that 

dx       dy 

*  This  is  the  manner  in  M^hich  analysts  have  agreed  to  express  an  isolated  ne- 
gative quantity  ;  which  must  necessarily  have  resulted  from  the  subtraction  of  a 
greater  from  a  less  quantity.  It  is  not,  however,  to  be  inferred  that  a  negative 
quantity  is  less  than  zero,  as  the  above  expression  indicates,  as  such  supposition 
would  be  manifestly  absurd,  Ed. 


THE  DIFFERENTIAL  CALCULUS.  95 

which  cannot  be  for  both  the  signs  ±  unless 

*=0,^  =  0....(l). 
ax:  ay 

By  continuing  to  imitate  the  reasoning  in  (49),  we  find  that  these 
same  conditions  must  exist  for  all  the  values  of  the  variables  that  ren- 
der the  function  a  minimum.  '    , 

Hence  (49),  we  have,  in  the  case  of  a  nuiuHatnn,  the  condition 

^  ^da^  dx  dy        dif 

and  in  the  case  of  a  minimum, 

d-z  d-z      ,    d^z  -,,,„,    o 

SO  that,  supposing  these  first  terms  do  not  vanish  for  the  values  of  j: 
and  y  given  by  (1),  the  condition  of  maximum  is 

and  the  condition  of  minimum, 

d-z  d'z         d'z 

'-dx^  dy  dx       dy^ 

In  either  case,  therefore,  the  expression  within  the  brackets  must 
have  the  same  sign  independently  of  the  sign  of  the  middle  term.  To 
determine  upon  what  other  condition  this  depends,  let  us  represent 
the  expression  by 

A  ±  2B  +  CorA(l  ±  2^  +  ^). 

B^      B^ 

Adding  —  —  —  =  0  to  the  quantity  within  the  parenthesis,  its 

JO.  A. 

form  is 

B  ,    .     C       B2 

A((i±  _)=+___). 

Now  this  expression  will  always  have  the  same  sign  as  A  provided 
C       B^ 
C  has,  and  that  "x  >  Ti'  *^^*  ^^'  ^^  7  ^^  °^  AC  —  B^  7  0,  be- 
cause then  the  factor  of  A  will  be  necessarily  positive.     Hence,  be- 
side (1),  the  condition  that  a  maximum  or  a  minimum  may  exist  is 


96  THE  DIFFERENTIAL  CALCULUS. 

and  we  are  to  distinguish  the  maximum  frcm  the  minimum  by  ascer-- 
taining  whether  the  proposed  values  of  x  and  y  render 

^ZOor/O, 

or,  which  amounts  to  the  same,  whether 

^ZOor/O. 

d?z  <Pz 

smce  -J—  and  -j-j-  have  the  same  sign. 

Should  any  of  the  values  determined  from  (1)  cause  the  coefficient 
of  h''  to  vanish,  there  will  be  no  maximum  or  minimum  for  those  va- 
lues unless  the  coefficient  of  the  following  term  vanishes  also. 


EXAMPLES. 

(65.)  1.  To  determine  the  shortest  distance  between  two  straight 
lines  situated  in  space. 

Let  the  equations  of  the  two  lines  be 

X  =  az  +   a)        3   i  x'  =  a'£  +   a!  ,,.. 

j,  =  6.+  ^P^M2/'  =  ^'-'  +  /3 ('^ 

then  the  expression  for  the  distance  between  any  two  points  (x,  i/,  z), 
{x'  y'  z')  is  {Anal  Geom.) 
j)^  =  u  =  {x  —  x'Y  +  (y  —  y'Y  +  {z  —  z'Y 

=  (a_a'  +  az— a's')'+(/3— iS'  +  bz—b'zy+{z—zy 
and  this  expression,  containing  the  two  independent  variables  z,  z'  is 
to  be  a  minimum.     Hence  by  the  condition  (1) 

—  =2(z-z')-\-2a{cc-.a'-{-az--a'z')-\-2b{(3-.^'+bz-.b'z')  =  o) 

dz       ^  V .  (2) 

^=_2(z_z')+2a'(a-a'+az— a'2')+26'(/3-/3'+6«-6'2') =0  N 
dz  -^ 

and  from  these  equations  the  proper  values  of  z,  z'  may  be  readily 
determined,  which  substituted  in  the  expression  for  u,  render  it  the 
least  possible.  That  these  values  really  belong  to  a  minimum  is  evi- 
dent, because. 


THE    DIFFERENTIAL    CALCULUS.  97 

=  _2  (1  4-aa'+  66'). 

«nd  this  proves  that-r-r,  ^— -  are  both  positive,  and  that 

'^  dsr    df- 

C^u     d^u  d^z 

d^'d£^~^d^/        ' 
Since  the  equations  of  a  straight  hne  passing  through  two  points 

{x,  y,  z),  {x',  y',  z')  are 

X  —  x'  =  a"  {z  —  z')  I 

y-y'  =  b"{z-z')i 

we  have,  by  substitution,  when  these  points  are  on  the  Hnes  (1) 

a  —  a'  +  az  —  a'  z'  =  a"  {z  —  z')  )  ,„v 

l3  —  ^'+b2_b'z'  =  b"{z  —  z')i-    '   '    '    ^"^^ 

hence,  if  this  Hne  be  that  in  question,  we  have,  by  combining  the 
equations  (2)  with  these,  the  conditions 

1  +  aa"  +  bb"  =  0,  1  +  a'  a"  +  b'  b"  =  0, 
which  conditions  shew,  that  this  minimum  straight  line  is  perpendicu- 
lar to  both  the  lines  (1).  (See  Anal.  Geom.)  From  these  conditions 
we  get 

ab  —  ab  a  b  —  ab 

by  means  of  which,  and  the  equations  (3),  the  expiession  for  D  be- 
comes 

^  =  ll-ll  V («  -  «')^  +  (6  -  b'r  +  {a'b  -  aby 

in  which,  if  we  substitute  the  value  of «  —  z  deduced  from  (2),  we 
obtain,  finally, 

(6  —  6')  (g  --•  a')  —  (a  —  a')  (,5  —  13') 
^  "^     V  (a  —  ay  +  (6  —  b'y  +  {a'b  —  ab'f" 

If  the  numerator  of  this  expression  vanish,  we  shall  have  D  =  0  ; 
so  that,  in  this  case,  the  lines  will  intersect.     Indeed,  the  condition 
of  intersection  of  the  two  lines,  (1),  we  know  {Anal.  Geom.)  to  be 
(6  _  b')  (a  —  a')  =  (a  —  a')  (/3  —  /3'). 
13 


98  THE  DIFFERENTIAL  CALCULUS. 

2.  Among  all  rectangular  prisms  to  determine  that  which,  having 
a  given  volume,  shall  have  the  least  possible  surface. 

Representing  the  three  contiguous  edges  of  the  prism  by  x,  y,  2, 
and  the  volume  by  a^  we  have 

u  =  2xij  +  2xs  +  2rjz  =  minimum, 
but  since 

«^ 

xyz  =  or  .'.  z  =  — 
xy 

2ft''        2tt'' 
.'.  u  =  2xt/  + j- =  muiimum. 

y         ^  . 

therefore  we  must  have  the  conditions 


that  is. 


du  _      du  _ 
dx  dij 

2a^  2a? 

2y  =  -^=0,2x-—  =  0 


from  which  we  obtain 

y  =  X  =  a  .'.  z  =^  a. 

If  these  values  really  correspond  to  a  minimum  they  must  fulfil 
the  conditions 

and  these  conditions  are  fulfilled,  since 

r^i  =  4  r— 1  =  4  r— 1  =  2 

■-da^  -■       ^'  ^df  J        ^'  ^dxdy^        ^' 

Hence  the  required  prism  must  be  a  cube. 

In  the  preceding  example  we  might  have  concluded,  without  re- 
curring to  these  conditions,  that  the  results  obtained  belong  to  the  re- 
quired minimum,  there  being  obviously  no  other  maximum  or  mini- 
mum, except  that  which  belongs  to  x  =  0,  1/  =  0,  z  =  oo ,  these 
being  the  values  which  cause  the  differential  coefficients  to  become 
infinite,  (see  art.  50.) 

3.  To  divide  a  given  number,  a,  into  three  parts,  such  that  the 
continued  product  of  the  mth  power  of  the  first  part,  the  nth  power  of 
the  second  part,  and  the  ^th  power  of  the  third,  may  be  the  greatest 
possible. 


I 


THE  DIFFERENTIAL  CALCULUS.  99 

™,     ^,  ^  ma  na  pa 

L  he  three  parts  are ; ; — , -, ; — , ~ — ; —    So 

m-\-n  +  pm-\-n-i-pm-\-n  +  p 

that  the  three  parts  are  to  each  other  as  the  exponents  of  the  proposed 
powers. 

4.  To  determine  the  greatest  triangle  that  can  be  enclosed  by  a 
given  perimeter. 

•    The  triangle  must  be  equilaterad.* 

On  changing  the  independent  variable. 

(66.)  It  is  frequently  requisite  to  employ  the  differential  coefficients 

dy   d^y    ....  .  ... 

-J-,  -T^  &c.,  m  which  X  is  considered  as  the  principal  variable  under 

a  change  of  hypothesis,  x,  and  consequently  y  being  assumed  as  a 
function  of  some  new  variable  /. 

It  is  therefore  of  consequence  to  ascertain  what  changes  take  place 
in  the  expressions  for  these  coefficients  in  such  cases.  This  we  may 
do  as  follows : 

Since  according  to  the  new  hypothesis 

y  =^Yx  and  x  =ft 
therefore  (33) 

dy  dy     dx       dy  dy    .    dx  (dy) 

dt        dx'  dt  '  '  dx       dt    '    dt        {dx) ' 

dti  dx 

where  for  brevity  {dy)  is  put  for  -^  and  {dx)  for  --. 

^y  _  d^y    ^V  ^y  ^^-^ 

IF  ~d^  '  d('       'di'l^ 

dx'        ^dl'         {dx)      ^       J        ^     ^  (^clxf 

In  a  similar  manner  we  might,  if  necessary,  find  the  expression  for 

dhi 

— J-.     It  appears,  therefore,  that 

^y   _  (%) 


dx         {dx) 

*  For  more  examples  the  student  may  refer  to  Jephsmi's  Fluxional  Calculus,  to 
Gamkr^s  Cdcul  Differentiel,  or  to  PuissanVs  Problimes  de  G6om6tric. 


J I     fir^       cVv       f-r  c 


100  THE  DIFFERENTIAL  CALCULUS. 

dx''  {dxf 

&c.  &c. 

Ift  =  y  the  hypothesis  requires  that  y  be  considered  as  the  princi- 
pal and  X  as  the  dependent  variable.     In  this  case 

(dy)  =  ^  =  1'  (dhj)  =  0,  &c.  {dx)  =  ^,  {d^x)  =-^,  &c. 


d,J 

_  1 

~  dx 

_      1 

dx 

{dx) 

dy 

d'y 

. 

{dr^x) 

dx"  {dxf 

&c.  &c. 

These  formulas  will  be  brought  into  use  in  the  second  section^ 


OHAPTZSR   IS. 


ON  THE  CASES  IN  WHICH  TAYLOR'S  THEOREM 
.  FAILS. 

(67.)  It  has  been  shown,  in  Chapter  II.,  that  the  general  devel- 
opment of  the  function  F  (x  +  h)  always  proceeds  according  to  the 
ascending  positive  powers  of^,  and  the  principle  upon  which  this 
fact  has  been  established  is  this :  viz.  that  Fx  and  F  (x  +  A)  must 
necessarily  contain  the  same  number  of  values  ;  or  in  other  words, 
the  same  radicals  that  enter  Fa?  must  also  enter  F(a?+/i)  but  no  others. 
Hence  we  might  extend  the  proposition  established  in  (4), and  say,  that 
not  only  the  general  development  proceeds  according  to  the  increas- 
ing positive  powers  of  h,  but  also  every  particular  development,  pro- 
vided the  particular  value  F  (a  4- A)  contain  the  same  radicals  as 
Fa,  and  no  more  ;  and  provided,  moreover,  that  Fa  is  not  infinite, 
which  we  have  seen  it  must  be  for  the  true  development  of  F  (o  -1-  h) 
to  contain  a  negative  power  of  h,  or  a  log.  h,  a  cot.  h,  &c. 

(68.)  As  F  (a;  +  h)  must  contain  the  same  radicals  as  Fxand  no 


THE  DIFFERENTIAL  CALCULUS.  101 

Others,  it  follows  that  F  {x  +  h)  —  Fx,  must  contain  the  same  radi- 
cals as  Fa:.  Now,  as  multiplying  or  dividing  an  expression  by  any 
rational  quantity,  can  neither  introduce  nor  destroy  radicals  in  that 

.      ^    F  ix-{-  h)  —  Fx    .  ^ 
expression,  we  infer  that  the  expression  for ,  h  be- 
ing rational,  must  contain  the  very  same  radicals  as  Fx,  and  no  others, 
whatever  be  the  value  of  h ;  but  when  /i  =  0 
F{x-\-h)  —  Fx  _  dFx 
h                      dx 
hence  the  first  differential  coefficient  must  contain  the  same  radicals 
as  the  function  Fx,  and,  by  the  same  reasoning,  the  second  differen- 
tial  coefficient  must  contain  the  same  radicals  as  the  first ;  conse- 
quently the  same  radicals  must  enter  each  differential  coefficient  that 
enter  into  the  original  function.     If,  therefore,  there  be  given  to  x  such 
a  particular  value  a,  that  any  one  of  the  expressions  that  may  be 
under  radicals  in  Fx  may  become  0,  that  radical  will  of  course  vanish 
from  the  function,  and  consequently  from  its  differential  coefficients. 
But  if  a  +  /«■  be  substituted  for  x  instead  of  a,  the  same  radical  will 
necessarily  be  preserved  in  F  (a  +  /i),  although  it  will  still  vanish 
from  Fa  and  the  differential  coefficients.     It  follows,  therefore,  that 
F  (a  +  h)  will  have  more  values  than 

so  that  this  cannot  be  the  true  development  of  F  (a  -f  h).  It  is  easy 
to  explain  why,  in  such  cases  as  this,  one  of  the  coefficients  and  in- 
deed all  that  follow  this,  must  become  infinite,  for  x  ^  a.  For  the 
exponent  of  the  radical  which  vanishes  for  this  value,  is  diminished 
by  unity  at  each  differentiation,  and  being  fractional  the  expres- 
sion under  it  will  at  length  appear  with  a  negative  exponent,  and  will 
continue  to  have  a  negative  exponent  in  all  the  succeeding  coeffi- 
cients ;*  so  that  these,  when  a  is  put  for  x,  become  infinite.  We 
have  observed  above,  that  the  failing  cases  of  which  we  are  speakings 

*  This  does  not  invalidate  the  previous  assertion,  that  the  same  radicals  enter 
the  coefficients,  that  appear  in  the  original  function  ;  for  the  radical  still  remains^ 
however  we  increase  or  diminish  the  fractional  exponent  by  integers,  for 


102  THE    DIFFERENTIAL    CALCULUS. 

arise  from  the  circumstance  of  x  =  a  causing  an  expression  to  disap- 
pear, which  is  under  a  radical  in  Fx.  The  student  must  not  confound 
this  disappearance  of  a  radical,  with  that  which  may  arise  from  a/ac- 
to7'  by  which  it  is  multiplied  becoming  0  for  x  =  a,  for  though  a 
radical  may  disappear  in  this  way  from  F^,  it  will  not  disappear  from 
all  the  differential  coefficients,  and  therefore  Taylor's  development 
will  hold.  Thus  if  a  radical  in  Fx  is  multiplied  by  (x  —  a)"",  m 
being  a  positive  whole  number,  this  radical  will  disappear  when  x  =  a, 
but  in  the  mth  differential  coefficient  the  factor  will  be  (x  —  a)"*""*, 
which  does  not  vanish  when  x  —  a,  but  becomes  =  1,  and  thus  the 
radical  with  which  it  is  connected  will  appear  in  this  coefficient. 

We  conclude,  therefore,  that  there  are  but  two  classes  of  values 
for  which  Taylor's  development  fails;  1°  those  which,  put  for  x, 
render  Fx  =  cc;    that  is,  those  which  are  roots  of  the  equation 

-:=r-  —  0  •  aJid  2°  those  which  substituted  for  x,  cause  an  expression 

under  a  radical,  to  vanish  from  Fx,  and  not  from  F  (x  +  h).  To 
this  latter  class  belongs  the  value  a?  =  a  for  every  function  containing 

\/x  —  a,  for  the  value  x  =  a  causes  this  radical  to  vanish  from  Fx, 

but  in  F  (a;  +  h)  it  enters  as  y/h. 

(69.)  In  order  to  examine  these  cases  more  completely,  let  in 

general 

1 

F{a  +  h)  ^A  +  B/i+C/i^+D/iH M/i"+N/i"'^^  +  &c.  .(1) 

represent  the  true  development  of  F  (ar  +  h)  for  x  =  a,  in  which 

»  +  -  denotes  a  fraction  falling  between  the  numbers  n  and  n  +  1. 

P 
We  shall  show  that  the  n  +  1th  differential  coefficient  derived  from 

the  function  Fx  becomes  infinite  for  x  =  a,  as  also  all  that  follow 
this,  but  the  preceding  coefficients  are  all  finite. 

Since  in  the  development  (1),  h  has  no  fixed  value,  we  may  dif- 
ferentiate relatively  to  /i,  and  we  shall  have, 

=B  +  2C/i-l-3D/iH +M*/i"-'+Ni/i"'*'p-'+  &c. 


dh 

^Ii?^=2C  +  2  •  3D/1+ +M,  /i'-HN2^"+^-'+  &c. 

&c.  &c. 

*  For  brevity  M,  will  be  here  used  to  denote  Uie  coefficient  ofh"-^  in  the  nth 
differential  coefficient  derived  from  M/j". 


THE  DIFFERENTIAL  CALCULUS.  103 

Now,  by  (30),  these  several  differential  coefficients  are  the  same  as 

dF{x  +  h)      ^dW  {X  +  h)^ 

the  brackets  denoting  the  values  when  x  =  a. 
Hence,  by  substitution,  in  the  foregoing  equations, 


1 


dF{x-{-h)    ^^^  2C/i  +  3Bh'  + m.Ji"-'  +  ^.h^^J-' 

^       ax 


+  &c (2) 


l^^-]=2C  +  2  •  3D/i  + M,h"-'  +  ^V^7-' 


+  &c (3) 

&c.  &c. 

Putting,  now,  ^  =  0  in  the  equations  (1),  (2),  (3),  &c.  we  have 
the  following  results. 

Fa  =  A 


dx" 
dx 


[^]  =  2.3D 


r:^]  =  M,./i"-"  +  N„0''-^  +  &c.  =  M„ 
"-  dx" 

&c.  &c. 

all  the  succeeding  differential  coefficients  being  obviously  infinite, 

because  the  exponent — -  1,  which  is  already  negative,  continually 

diminishes  by  unity. 

(70.)  It  follows,  therefore,  that  if  the  true  development  of  F(a:  -\-h), 
arranged  according  to  the  increasing  exponents  of  A,  contain  for 
X  =  aa.  fractional  power  of  /i,  comprised  between  the  powers  h"  and 


104  THE  DIFFERENTIAL  CALCULUS. 

h"'*'\  then  the  several  terms  of  this  development  will  be  correctly  de- 
termined by  Taylor's  theorem,  as  far  as  the  term  containing  h"  in- 
clusively ;  but  the  terms  beyond  this  become  infinite,  and  therefore 
do  not  belong  to  the  true  development. 

If  a  tefm,  in  the  true  development  of  F(a  +  h),  contain  a  negative 
power  of /i,  this  should  be  the  leading  term,  as  the  arrangement  is 
according  to  the  increasing  exponents ;  therefore,  this  first  term, 
when  X  =  a  and  h  =  0,  must  be  infinite,  and  consequently  all  the 
differential  coefficients,  (2),  (3),  &c.  must  be  infinite. 

(71.)  The  converse  of  these  inferences  are  true,  viz.  1°.     If,  in 

the  general  development  of  F(x  +  h),  the  coefficient  is  the 

first  which  becomes  infinite  for  a  particular  value  of  x,  then,  in  the 
true  development,  arranged  according  to  the  increasing  exponents  of 
h,  the  term  immediately  succeeding  that  which  contaii.s  h",  will  con- 
tain a  fractional  power  of  h,  the  exponent  being  between  n  and  n-\-l. 
For  it  is  obvious,  that  in  order  that  the  n  +  1th  may  be  the  first  of 
the  coefficients  (2),  (3),  &c.  which  contain  a  negative  power  of /i, 

h"'*'p  must  be  the  first  fractional  power  of  h  which  enters  the  develop- 
ment (1),  »  +  -  being  between  n  and  n  +  I.  2°  If,  for  a  particu- 
lar value  of  X,  the  function  Fx  become  infinite  then  will  all  the  dif- 
ferential coefficients  become  also  infinite,  and  the  true  development 
will  contain  a  negative  power  of  h,  or  else  a  log.  h,  a  cot.  h,  &c. 
For,  if  the  true  development  of  F(a  +  h)  did  not  contain  a  negative 
power  of  ^,  nor  a  log.  /»,  a  cot.  h,  &c.  Fa,  which  this  becomes  when 
h  =  0,  could  not  be  infinite  ;  hence,  such  a  function  of  h  must  enter» 
and  therefore,  as  shown  above,  all  the  differential  coefficients  become 
infinite,  for  x  =  a.  It  is,  therefore,  necessary  to  examine  the  func- 
tion Fa,  before  we  deduce  the  coefficients  from  Fx. 

(72.)  To  obtain  the  true  development  of  the  function  for  those, 
particular  values  of  the  variable,  which  cause  Taylor's  theorem  to 
fail,  the  usual  course  is  to  recur  to  the  ordinary  process  of  common! 
algebra,  after  having  substituted  a  -\-  h£or  xin  Fx. 

Suppose,  for  example,  the  function  were 


Fa:  =  2ax  —  x^  -\-  a  V  ^  —  aP, 
and  that  we  required  the  development  of  ■F(ar  -f  h),  for  x 


THE    DrPFERENTIAL   CALCULUS.  105 

Taking  the  differential  coefficient,  we  have 

&c.  &c. 

As,  therefore,  the  first  differential  coefficient  becomes  infinite  for 
the  proposed  value  of  x,  we  conclude  that  the  true  development  of 
the  function  for  that  value,  when  arranged  according  to  the  increasing 
exponents  of /j,  has  a  fractional  power  of /t  in  the  second  term,  the 
exponent  of  this  power  being  between  0  and  1. 

Substituting,  then,  a -\-  ^  for  a^  in  Fx,  we  have 


F(a  -\-  h)  =  a'' —  h"  -{-  a  V2ak  +  k' 
=  aP  —  h^  +  ah^  (2a  +  h)^ 
Developing  (2a  +  h)^  by  the  binomial  theorem,  we  have 

(2a  +  hf  =  {2af  +  — ^ ^  +  &c. 

2  (2a)  2        8  (2a)  3 

consequently, 

S  5 

F{a  +  h)=a^+  a  {2a)h^  +  —^ h? ^  +  &c. 

2  (2a)-2  8  (2a)t 

Again  let  the  function  be 

Fx  —  \/  ar  +  (.r  —  of  log.  {x  —  a), 
and  let  it  be  required  to  find  the  development  of  F(a  +  h).     Here 
Fa  =  \/a  +  0  X  (30. 

It  becomes  necessary,  therefore,  first  to  ascertain  whether  this  ex- 
pression is  infinite ;  for,  if  it  be,  we  are  not  to  proceed  with  the  differ- 
entiation, but  to  infer,  agreeably  to  art.  (70),  that  the  proposed  func- 
tion, and  all  the  differential  coefficients,  become  infinite  for  a;  =  a 
and  that  consequently  the  true  development  contains  either  a  nega- 
tive power  of  ft,  or  a  logarithm  of  h.  Now,  by  the  method  explained 
in  (44),  we  find  that,  when  x  =  a,  the  true  value  of 

{x  —  ay  log.  (a?  —  a)  =  — ^ — - — 


log.  {x  —  a) 
is  uifinite.  Hence  the  development  of  F (a  -\-  h)  contains  log.  A,  for 

14 


105  THE  DIPFERENTIAI,  CALCULUff. 

we  readily  see  that  no  negative  power  of  A  can  enter.     Substituting 
o  +  &  for  X,  in  Fx,  we  have 

F(o  +  /i)  =  (a  +  ft)  2  +  ^2  Iq„  i^ji, 

SCHOLIUM. 

(73.)  In  the  preceding  remarks  on  the  development  of  functions 
for  particular  values  of  the  variable,  tVe  have  said  nothing  about  the 
values  of  A,  the  increment  of  that  variable,  having  indeed  considered 
that  increment  as  indeterminate,.^or  rather  of  arbitrary  value.  It 
must,  however,  be  observed  that,  although  the  particular  value  which 
we  give  to  x  does  not,  in  any  case,  fix  the  value  of  /i,  it  may  neverthe- 
less fix  the  limit  between  which  and  0  all  the<  values  given  to  h  must 
be  comprised,  in  order  that  for  particular  values  of  x,  Taylor's  de- 
velopment may  not  fail.  This  fact  is  very  plain,  for  if  the  develop- 
ment holds  for  all  values  of  x  from  ar  =  a  up  to  a;  =  6,  but  fails  for 

*  The  above  example  is  from  Lagrange,  {Calcid  des  Fonctions,  p.  75,)  who  has 
given  a  faulty  solution  of  it,  which  however  is  copied  by  Gamier  and  other  writers 
on  the  Calculus.     The  solution  here  objected  to  is  as  follows ; 

"  Soit 

fx=-\/x-{-(,x  —  a)*log.  (x  —  a) 
on  auia  ces  fonctions  d6riv6es 

fx  =  — h  2  (x  —  a)  log.  (x  —  a)  -{-  x  —  a 

•'  8xVa?      a;— o 

&c. 
Si  on  fait  x  =  a,  lafonction  seconde/"x  devient  infine,  ainsi  que  toutes  les  sui" 
vantes. 

"Ainsi  le  d^veloppment  de/(x  +  h)  par  la  formule  gifn6rale  deviendra  fautif 
dans  le  cas  de  x  =  a,  e(  iZ  contiendra  nicessairement  le  terme  h^  log.  A." 

This  solution  is  faulty,  inasmuch  as  it  is  assumed  that/"x  is  the  first  derived 
function  that  becomes  infinite  for  x  =  a,  whereas /'x  and/x  are  also  infinite ;  but 
a  greater  fault  is,  that  this  process  does  not  lead  to  the  true  conclusion,  for  the 
inference  in  italics  does  not  follow  from  it,  but  this,  viz.  that  the  sought  develop, 
ment  contains  neither  log.  of  A  nor  a  negative  power  of  A,  but  it  contains  a  frac- 
tional power  of  A,  the  exponent  being  between  1  and  2,  which  is  not  the  true  con- 
clusion. 


THE  DIFFERENTIAL  CALCULUS.  107 

JC  =  6,  then  will  the  development  hold  when  a  -^  his  substituted  for 
ar  in  Fx,  provided  h  be  taken  between  the  limits  h  =  0  and  h  =  b  —  a, 
or  more  strictly,  provided  it  does  not  exceed  these  limits.  In  like 
manner,  if  the  development  hold  for  all  values  of  .r  from  x  =  b  down 
to  a:  =  «,  but  fails  for  x  =  a,  then  will  the  development  hold  when 
b  —  his  substituted  for  x  for  all  values  of  h  from  h  =  0,  io  h  =  b  —  a, 
but  it  will  not  hold  for  the  value  of  h  immediately  succeeding  this 
last ;  and  it  is  obvious  that  h  will  always  be  subject  to  such  restrictions 
unless  the  development  holds,  not  merely  for  x  =  a,  but  universally. 
When,  therefore,  we  find  that  for  x  =  a  the  differential  coefficients 
do  not  any  of  them  become  infinite,  all  that  we  can  conclude  is  that 
the  development  of  F  (a  it  h)  is  according  to  Taylor's  theorem  for 
all  values  of  h  between  some  certain  finite  value  h',  which  may  in- 
deed be  indefinitely  small,  and  0,  and  it  is  only  when  this  is  not  the 
■case  that  the  theorem  is  said  by  analysts  to  fail.  We  have  thougl.t 
it  necessary  to  point  out  these  circumstances  to  the  student,  seeing 
that  some  authors,  from  not  attending  to  them,  have  fallen  into  very 
important  errors,  and  have  laid  down  erroneous  doctrines  with  respect 
to  the  failing  cases  of  Taylor's  theorem.  Thus  Mr.  Jephson  at  page 
191,  vol.  i.  of  his  Fluxionial  Calculus,  a  work  containing  much  valua- 
ble information,  says  "  It  may  further  be  observed  that  Taylor's  theo- 
rem always  fails  when  the  assigned  value  of  a?  causes  any  of  the  terrrs 
to  become  imaginary,  and  that  this  may  take  place  without  causing 

the  function  itself  to  be  imaginary ;  thus  take /a?  =  c  +  x^  \/x  —  a 
if  we  suppose  x  =  0,fx  =  c,f'x=  O^hnt  f"x,f"'x  ....  all  con- 
tain V  —  a."  From  this  it  would  appear  that  Taylor's  theorem 
may  fail  to  give  the  true  development  in  other  cases  besides  those 
which  cause  the  differential  coefficients  to  become  infinite,  which, 
however,  is  not  true.  Whenever,  for  any  particular  value  of  x,  Tay- 
lor's coefficients  become  imaginary,  we  must  infer,  agreeably  to  the 
statement  in  (4),  that  the  function  F(a?  +  h)  becomes  imaginary  for 
that  value  of  x;  fe  being  of  course  limited  as  above  explained.  In 
the  example  just  quoted,  where 


f'x-=^  2x  \/ X  —  a  +      , 

''  V  X  —  a 

2x                     Ix* 

f'x  =  2    Vx  —  a  +  "7=        

"^  V  X  —  a 

&c.  &c. 


(x—  af 


108  THE  DIFFERENTIAL  CALCULUS. 

the  function  /(x  +  h)  becomes,  when  x  =  0,/(0  -{■  h)  =  c  +  h!^ 
^/  /i  —  a  and  the  development  is 


c  +  /i^  V/i  —  a  =  c  +  O/i  +  ^  —alr-'i'  ^   , F  +  &c. 

2v  —  a 

and  this  is  the  true  development,  for  h  must  not  exceed  the  limits 
h  =  0,  and  h  =  a,  since  a;  =  a  causes  the  differential  coefficients  to 
become  infinite,  and  therefore  the  development  to  fail. 

With  regard  to  the  faihng  cases  of  Maclaurin's  theorem,  it  may  be 
observed  that  they  are  very  diflerent  from  the  failing  cases  of  Taylor's. 
Whatever  be  the  form  of  the  proposed  function,  its  general  develop- 
ment, according  to  Taylor's  theorem,  never  fails  ;  but  the  failure  of 
Maclaurin's  theorem  always  arises  from  the  form  of  the  proposed 
function  and  it  is  the  general  development  that  fails,  and  consequently 
all  the  particular  cases.  For  it  is  obvious  that  every  function  or  any 
of  its  differential  coefficients  which  become  infinite  when  x  =  0,  will 
fail  to  be  developable  by  Maclaurin's  theorem. 

Before  terminating  these  remarks  it  may  be  proper  to  observe  that 
the  student  is  not  to  attribute  what  analysts  have  been  pleased  to  term 
the  failing  cases  of  these  theorems  to  any  defect  in  the  theorems 
themselves  ;  on  the  contrary  they  would  be  very  defective  if  they  did 
not  exhibit  such  cases.  All  that  is  meant  is,  that  the  function  in  par- 
ticular stales  may  fail  to  be  developable  according  to  Taylor's  series^ 
and  under  particular  forms  it  may  fail  to  be  developable  according  to 
Maclaurin's  series  ;  so  that,  in  fact,  these  theorems  fail  to  give  the 
true  development  only  when  that  development  is  impossible. 

(74. )  Let  us  now  examine  implicit  functions,  and  let  us  suppose 
that  X  =  a  causes  a  radical  to  vanish  from  ¥x  in  consequence  of  a 
factor  of  it  vanishing ;  we  have  seen  (68)  that  such  radical  will  reap- 
pear in  some  of  the  differential  coefficients,  suppose  it  appears  in  the 
first  which  requires  that  the  factor  spoken  of  be  x  —  a,  then  for  x  =  a 
this  coefficient  will  have  more  values  than  the  proposed  function,  as 
it  contains  a  radical  more.  But  if  the  function  that  Fx  or  y  is  of  a? 
be  only  implicitly  given,  that  is  by  means  of  an  equation  without  radi- 

cals,  we  know  (52)  that  the  expression  for  -^  will  be  also  without  radi- 
cals and  from  such  an  expression  it  does  not  at  first  seem  clear  how 


THE  DIFFERENTIAL  CALCULUS.  109 

we  are  to  deduce  the  multiple  values  alluded  to,  and  which  might  be 
obtained  by  solving  the  equation  for  y  and  thus  introducing  the  radi- 

cal.     But  since  the  expression  for  -~  appears  under  the  form  of  a 

fraction,  viz.  (52) 

du 

dif  dx 

di'^~  dii'   •  '  '  (^^' 

dy 

we  readily  perceive  that  one  case  is  possible,  and  only  one,  in  which 

this  fraction  may  take  a  multiplicity  of  values  besides  those  implied-in 

y,  viz.  the  case  in  which  it  becomes  f ;  that  is,  when  the  following 

conditions  exist  simultaneously,  viz. 

du       „  du      •  ,„^    ' 

^=0,^  =  0....  (2), 

so  that  these  conditions  are  those  which  must  exist  for  every  value  of 

4r  which  destroys  a  radical  in  Fx  but  not  in  ~. 

(75.)  Whenever,  therefore,  any  particular  value  of  x  destroys  a 

radical  in  y,  but  not  in  -p,  then  the  expression  (1)  must  take  the  form 

•g,  and  admit  of  the  necessary  multiple  values. 

The  rules  laid  down  in  (41)  enable  us  to  determine  the  true  value 
of  the  fraction  (1 )  in  the  proposed  circumstances,  that  is  when  it  takes 
the  form  »,  for  there  is  but  one  independent  variable,  viz.  x.  By  dif- 
ferentiating numerator  and  denominator  separately,  we  have,  by  the 
article  referred  to, 

dru  d^u      dy 

r-dy  dx^        dxdy  '  dx 

^di^^  ~t  du  d'u      ^^  •  •  •  •   (3); 


"hence. 


dxdy         dif   '  dx 


r—  4-  2    -^    f!^  4-  ^  r^^2^  _  n 
^dx"    ^       dxdy  '  dx       df  W  J 


This  being  a  quadratic  equation  furnishes  two  values  for  [— ], 


'dx- 


110  THE  DIFFERENTIAL  CALCULUS. 

which  are  all  that  belong  to  the  fraction  (1)  or  to  [-p]  unless,  indeed, 

both  numerator  and  denominator  of  (3)  also  vanish  for  a  value  of  Tgiven 
by  the  conditions  (2),  in  which  case  we  must  differentiate  again  as  in 

art.  (41 ),  when  the  value  of  [-7-;]  will  be  given  by  an  equation  of  the 

third  degree  which  will  furnish  three  values,  and  so  on  ;  and  in  gene- 

dy 
ral,  if  the  fraction  or  [-7-]  admit  of  n  values,  the  equation  which  de- 
termines them  can  be  obtained  only  by  differentiating  n  times,  which 
will  lead  to  an  equation  of  the  nth  degree,  and  it  is  plain  that  the  ra- 

1 
dical  destroyed  in  Fx  must  be  of  the  -th  degree,  seeing  that  it  gives 

^y  .  . 

to  -j-  n  additional  values. 
ax 

Suppose,  for  example,  we  had  the  function 
y  =  x-\-  {x  —  a)  \/ X  —  6 
dy 


and  for  X  =  a 


<^                                  2-Jx  —  b 
dv  


dM 
so  that  the  radical  which  disappears  in  y  appears  in  -j- ;  this,  there- 
fore, has  two  values. 

Now  let  the  same  function  be  given  in  an  implicit  form  and  with- 
out radicals,  viz. 

«  =  {y  —  x)^  —  {x  —  of  {x  —  b)  =  0, 

rf/ft  du 

.'.-  =  ~2{y-x)-{x  —  a)  {3x  _  26  +  a),  -  =  2  {y-x). 

Since  for  x  =  a,  1/  =  o,  therefore  both  these  expressions  become 

0 ;  hence  [-p]  =,-.     Taking,  then,  the  diflferentials  of  both  numera- 

tor  and  denominator  of  the  fraction, 

2  (t/  —  a?)  +  (x  —  a)  (3x  —  2b  -\-  a)  _  0 
2{y—x)  0' 


THE  DIFFERENTIAL  CALCULUS.  Ill 

we  find  when  x  =  a  that  it  becomes 

^dx^  ^     dii 

therefore, 

as  before. 

(76.)  Suppose  now  that  the  radical  which  disappears  from  Fx  by 

reason  of  a  factor,  disappears  also  from  -^,  and  that  it  appears  in  -j^t 

which  is  the  same  as  supposing  that  the  vanishing  factor  is  (x  —  a)'. 

In  this  case  —  will  have  the  same  number  of  values  that  Fa;  has  for 
ax 

X  =  a,  but  additional  values  will  belong  to  -7^,  so  that  we  musthave 

cFy     _0 

and  the  true  values,  when  the  function  is  implicit,  will  all  be  deter- 
mined by  the  principles  already  employed. 
For  example  :  the  explicit  function 

y  =  X  +  (x  —  ay  V  X 
gives  when  x  =  a 

M  =  <..[|]=i,[§]  =  ±w«. 

and  we  shall  see  that  the  same  values  are  equally  given  by  the  imph- 
cit  function 

iy  —  ^Y  =  (^  —  «)*  ^. 

by  applying  the  foregoing  process  to  -7^. 

For  by  successively  differentiating,  and  representing,  for  brevity, 
the  several  coefficients  derived  from  y  by  p,  p'',  &c.  we  have 

2  (p'  —  1)  (y  —  x)  =  (x  —  ay  (5x  —  a)  .-.  [;?']  =  1 
(p  —  1)='  +  p"  {y^x)=2{x^  ay  {5x  —  2a) 


112  THE  DIFFERENTIAL  CALCULUS. 

.   r  „-,  ^  .2{x-aY{5x  —  2a)  —  (p'—iy     ^  0 
•  •  L'^  -J        L  y—x  J         0 

=  r6  {x  —  g)  (5ar  —  3a)  —  2p"  jp'  —  1)  -,   _  0 
L  p'  — 1  -I    ~  0 

^    60g  —  48g  —  2p"'  {y'  —  1)  —  2/>"3     _    12a  —  2  [p'"^} 
~  L  p"  J  5"^ 

...  3  [p"2]  =  12a.-.  [//']  =  ±  2  V  a, 
as  before. 

The  two  examples  following  will  suffice  to  exercise  the  student  in 
this  doctrine,  which  is  merely  an  extension  of  the  principles  treated 
in  Chapter  V.  to  implicit  functions. 
1.  Given 

f  =  {x-aY(x-^b) 

to  determine  the  values  of  -z-,  when  x  =  a, 
ax 

dii. 


2.  Given 

(^y  __  xf  —  {x  —  ay  {x  —  b)=0 

dy         iPi/ 
to  determine  the  values  of  -j-  and  -irr,  when  x  =  a. 

_dy  d^ii 


We  here  terminate  the  First  Section,  having  fully  considered  the 
various  particulars  relating  to  the  diiferentiation  of  functions  in  gene- 
ral. 


THE    DnFTERENTIAL    CALCULUS.  113 


SECTION   II. 

APPLICATION  OF  THE  DIFFERENTIAL  CALCULUS 
TO  THE  THEORY  OF  PLANE  CURVES. 


CBAFTSn  Z> 
ON  THE  METHOD  OF  TANGENTS. 

(77.)  We  now  proceed  to  apply  the  Calculus  to  Geometry,  and 
shall  first  explain  the  method  of  drawing  tangents  to  curves. 

The  general  equation  of  a  secant  passing  through  two  points  (x', 
y')i  {^"i  !/")>  i^^  ^"y  plane  curve,  is  {Anal.  Gcom.) 
V'  —  '/" 

y—y'  =  '^'^],"{^  —  ^')^ 

y'  —  J/",  being  the  increment  of  the  ordinate  or  proposed  function 
corresponding  to  x  —  x"  the  increment  of  the  abscissa  or  independ- 
ent variable.     The  limit  of  the  ratio  of  these  increments,  by  the  prin- 

ciples  of  the  calculus,  is  -y-, ;  that  is  to  say,  such  is  the  representation 

of  the  ratio  when  x   —  x"  =  0,  and,  consequently,  y'  —  y"  =  0. 
But  when  this  is  the  case  the  secant  becomes  a  tangent.     Hence  the 
equation  of  the  tangent,  through  any  point  (x',  y')  of  a  plane  curve,  is 
dy' 

y—y''='db'i''--^')  •  •  •  •  (!)• 

dii' 
It  appears,  therefore,  that  the  differential  coefficient  -i-,  for  any 

proposed  point  in  the  curve  has  for  its  value  the  trigonometrical  tan- 
gent of  the  angle  included  by  the  linear  tangent  and  the  axis  of  x,  that 
is,  provided  the  axes  are  rectangular.  If  the  axes  are  oblique,  the 
same  coefficient  represents  the  ratio  of  the  sines  of  the  inclinations 
of  the  linear  tangent  to  these  axes.     {See  Anal.  Geom.) 

By  means  of  the  general  equation  (1)  we  can  always  readily  de- 
termine the  equation  of  the  tangent  to  any  proposed  plane  curve  when 
the  equation  of  the  curve  is  given,  nothing  more  being  required  than 
to  determine  from  that  equation  the  differential  coefficient. 

15 


114  THE  DIFFERENTIAL  CAtCTTLUS. 


dy' 
for  the  ellipse.     We  here  have  to  determine  -—  from  the  equation 


Suppose,  for  example,  it  were  required  to  find  the  particular  form 

3re  have  to  determine  - 
d 

and  which  is 

dy'  _        BV 

dx'  ~        Ay 
therefore  the  equation  of  the  tangent  is 

y—y  =— Ay  ^''■~"''^* 

{x,  y)  being  any  point  m  the  curve,  and  (x,  y)  any  point  in  the  tan- 
gent. 

Again  ;  let  it  be  required  to  determine  the  general  equation  of  the 
tangent  to  a  line  of  the  second  order. 

By  differentiating  the  general  equation 

Ay"  +  Bx'tj'  +  Ca;'2  +  Dy'  +  Ex'  +  F  =  0,.  * 
we  have 

2Ay'^  +  Bar'  -^  +  By'  +  2Ca;'  +  D  -^  +  E  =  0^ 

dy'  _        2Ca;'  +  By'  +  E 
•*'  Ix'  2Ay'  +  Ba;'  -\-~I) 

so  that  the  general  equation  is 

2Cx'  +  By'  +  E 
y-y  =-2Ay'  +  Bx'+^^^-^^- 
(78.)  As  the  normal  is  always  perpendicular  to  the  tangent,  its 
general  equation  must  be,  from  (1), 

!/-2/'=-^C^-^')  t.-..(2). 

♦  The  general  equation  of  lines  of  the  second  order  in  its  most  commodiou* 

form  is 

t/'J  =:  mx'  +  nx'\ 

from  which,  by  differentiation,  we  have 

dy' m-{-2nx' 

'd^'~       2y' 
and  the  equation  of  the  tangent  to  a  line  of  the  second  order  is  thereforo 

,      m  +  2nx' 

dx' 
^  dy'^  '  Ed. 


THE  DIFFERENTIAL  CALCULUS. 


116 


It  is  easy  now  to  deduce  the  expressions  for  the  subtangent  and  sub- 
normal. For  if,  in  the  equation  of  the  tangent,  we  put  y  =  0,  the 
resulting  expression  for  x  —  x'  will  be  the  analytical  value  of  that  part 
of  the  axis  of  x  intercepted  between  the  tangent  and  the  ordinate  y'  of 
the  point  of  contact,  that  is  to  say,  it  will  be  the  value  of  the  subtan- 
gent T^  {Anal.  Geom.), 

.'.  T  =  — ^  ....  (3). 

dx' 
If,  instead  of  the  equation  of  the  tangent,  we  put  y  =  0  in  that  of  the 
normal,  then  the  resulting  expression  for  x  —  x'  ^vill  be  the  value  of 
the  intercept  between  the  normal  and  the  ordinate  y\  that  is,  it  will 
belong  to  the  subnormal  N,, 

.•.N,=,'|....(4). 


As  to  the  length  of  the  tangent  T,  since  T  =  -s/i/^  -f  T/,  w« 
have,  in  virtue  of  (3), 

dx'^ 


Also,  since  the  length  of  the  normal  N  is  N  =  V  y'''  +  N'^  w© 
have,  by  (4), 

N  =  !/'  V  (1  +   ;g[)  .  .  .  .   (6). 

The  foregoing  expressions  evidently  apply  to  any  plane  curve  what- 
ever, that  is,  to  any  curve  that  may  be  represented  by  an  equation 
between  two  variables,  whatever  be  its  degree,  or  however  compli- 
cated its  form. 

We  shall  now  give  a  few  examples  principally  illustrative  of  the 
method  of  drawing  tangents  to  the  higher  curves,  for  which  purpose 
we  shall  obviously  require  only  the  formula  (3),  for  it  is  plain  that  to 
any  point  in  a  curve  we  may  at  once  draw  a  tangent,  when  the  length 
and  position  of  the  subtangent  is  determined.  Or,  knowing  the  point 
{x',  y'),  we  may,  by  putting  successively  x  =  0  and  t/  =  0,  in  the  equa- 
tion (1), determine  the  two  points  in  which  the  required  tangent  ought 

dy' 
to  cut  the  axes  of  the  coordinates  and  then  draw  it  through  them.  If-—-. 

or 


116  THE  DIFFERENTIAli  CALCULUS. 

is  0  at  the  proposed  point,  the  tangent  will  be  parallel  to  the  axes  of  x, 

because,  as  remarked  above,  -r-,  is  the  value  of  the  trigonometrical 

ax 

tangent  of  the  inclination  of  the  linear  tangent  to  the  axes  of  x,  and  for 
this  reason  also  the  tangent  will  be  parallel  to  the  axes  of  y  when  — -, 
is  infinite  at  the  proposed  point. 


EXAMPLES. 

(79.)     1.  To  draw  a  tangent  to  a  given  point  P  in  the  common 
or  conical  parabola.  • 

By  the  equation  of  the  curve 

.^  ^y'  _  P 

'  '  'dx'        2y' 

'       -^        2y'  p 

Hence,  having  drawn  from  P,  the  perpendicular  ordinate  PM,  if  wo 
set  off  the  length,  MR,  on  the  axis  of  x,  equal  to  twice  AM,  and  then 
draw  the  line  RP,  it  will  be  the  tangent  required. 

2.  To  determuie  the  subtangent  and  subnormal  at  a  given 
point  (x'j  y')  in  the  parabola  of  the  nth  order,  represented  by  the 
equation 

y  =  aaf 

dy' 

ax 
.-.  T,  =  t/'  -r  nax"-^  =  -,  N,  =  y'  nax'"*-^  =  na'aP"-^  or  —-, 

3.  To  determine  the  subtangent  at  a  given  point  in  the  loga- 
rithmic curve. 

The  equation  of  this  curve  related  to  rectangular  coordinates  is 
y  =  a% 
which  shows  that  if  the  abscissas  x  be  taken  in  arithmetical  progres- 
sion, the  corresponding  ordinates  y  will  be  in  geometrical  progres- 
sion, so  that  the  ordinates  of  this  curve  will  represent  the  numbers, 
the  logarithms  of  which  are  represented  by  the  corresponding  ab- 


THE  DIFFERENTIAL  CALCULUS. 


117 


scissas,  a  being  the  assumed  base  of  the  system.  Hence,  calling  the 
modulus  of  this  base  m,  we  have,  by  differentiating  (13), 

dy  _  1 

drx~my' 


.-.  T,  =  y  - 


y 


Hence  the  remarkable  property  that  the  subtangent  is  constantly  equal 
to  the  modulus  of  the  system,  lohose  base  is  a. 

4.  To  determine  the  subtangent  at  a  given  point  in  the  curve 
whose  equation  is 

a^  —  3axy  +  y^  =  0. 


Here 


A=3^_3.,-3„|  +  34^  =  0. 


•**  dx' 
.-.  T,  = 


ay  —  X- 


y 


aa/* 


y^  —  ax  y 


ay  —  X  * 
6.  To  draw  a  tangent  to  a  rectangular  hyperbola  between  the 
asymptotes,  its  equation  being  xy  =  a. 

T,  =  x'. 

6.  To  determine  the  subtangent  at  a  given  point  in  a  curve 
whose  equation  is  y"'  =  aif,  which,  because  it  includes  the  common 
parabola,  is  said  to  represent  parabolas  of  all  orders. 

T^  =  ^x'. 
n 

7,  To  determine  the  subtangent  at  a  given  point  in  a  curve 
whose  equation  is  x"  y"  =  a,  which,  because  it  includes  the  com- 
mon hyperbola,  is  said  to  belong  to  hyperbolas  of  all  orders. 

T^  =  ^x'. 
m 

(80.)  If  the  proposed  curve  be  related  to  polar  coordinates,  then 
the  expressions  in  last  article  must  be  changed  into  functions  of  these. 

If  the  curve  AP  be  related  to  polar  coor- 
dinates FP  =  r,  PFX  =  CO,  then  if  PR  be 
a  tangent  at  any  point,  P  and  PN  the  nor- 
mal, and  if  RFN  be  perpendicular  to  the  ra- 
dius vector  FP,  the  part  FR  will  be  the  polar 
tubtangentt  and  the  part  FN  the  polar  sub- 


118  THE  DIFFERENTIAL  CALCULUS. 

normal.  When  the  pole  F  and  the  point  P  are  given,  it  is  obvious 
that  the  determination  of  the  subtangent  FR  will  enable  us  to  draw 
the  tangent  PR. 

The  formulas  for  transforming  the  equation  of  a  curve  from  rec- 
tangular to  polar  coordinates,  having  the  same  origin,  are  {Anal: 
Geom.) 

X  =  r  cos   u,y—'r  sin.  u,  x^  -\-  y'^  =  t'^, 
and  the  resulting  equation  of  the  curve  will  have  the  form 

r  =  Fw,  or  F  (r,  u)  =  0, 
in  which  we  shall  consider  w  as  the  independent  variable.     Now 
RF  =  PF  •  tan.  Z  P  =  **  tan.  Z  P,  but  by  trigonometry, 

tan.  cj  —  tan.  a 


tan.  Z  P 


1  +  tan.  w  tan.  a 


that  is,  since 


dy      .  w 

tan.  a  =  -y-  and  tan.  u  =  - 

ax  X 

tan.  Z  P  = "       f 

X       ax 

therefore  r  times  this  expression  is  the  value  of  the  polar  subtangent. 

dy 
But  the  differential  coefficient  -p,  which  implies  that  x  is  the  princi- 
pal variable,  ought  to  become,  when  the  variable  is  changed  to  Ur 
(66), 

dx  dy 

dv        dv          dx  -n  du  dui 

■r- =  rr- -^ -T- -'•  tan.  z  P  =  — r 7- 

dx        d(ji         aw  dx      .         du 

X  _i_  „  — — 

du  *'    dox 

Also,  from  the  above  formulas  of  transformation, 

dx         dr  .  dy         dr     . 

=  — —  COS.  u  —  r  sm.  w,  -^ —  =  -7—  sm.  w  +  r  cos.  w 

du         du  du         du 

•  *.  H  —, —  =  **  ~; —  sin.  w  cos.  u  —  r^  sin.^  u 
•^  acj  du 

^y  ^^     ■  L  ^        a 

X  -r—  =  5*  —, —  sm.  w  COS.  w  +  r*  cos."  w 
au  du 


■THE  DIFFERENTIAL  CALCULUS.  119 

dx  dr  „  o   • 

X =  r  -7—  COS.   w  —  r*  sm.  w  cos.  u 

dcjj  du 

dy  dr    .    .       ,     „   . 

«  — ^  =  r  -5 —  sin.^  '^  +  IT  sin.  w  cos.  u 


whence 


dx  dy  „        dx    .        dy  dr 

x-f-  =  —ir',x-r-+  y-r~—  ^  -T- 

dcj  aw  aw  aw  aw 


consequently, 

tan.  /  P  =  -^  .-.  RF  =  -7-  =  subtangent. 
dr  dr 

dw  dw 

Also 

FN  = =  r^  -T- = =  subnormal. 

FR  dr         dw 

dw 

(81.)  We  shall  apply  these  formulas  to  spirals,  a  class  of  curves 

always  best  represented  by  polar  equations. 

8.  To  determine  the  subtangent  at  any  point  in 

the  Logarithmic  Spiral,  its  equation  being 


r  = 

=  a 

dr 

w 

w 

— 

log. 

a  a  = 

dw 

m 

J 

.  a  a  =  -^—  I     /^ ^ 

m  I  / 

.'.  IT  —  -7 —  =  mr  =  i 
dw 

Hence,  if  a  represent  the  base  of  the  Napierian  system,  since  the 
modulus  will  be  1,  the  subtangent  will  be  equal  to  the  radius  vector, 
and  therefore  the  angle  P  equal  to  45°,  because  tan.  ^  P  —  1. 

Since,  by  the  equation  of  this  curve,  log.  r  =  w  log.  a,  it  follows 
that,  if  a  denote  the  base  of  any  system,  the  various  values  of  the 
angle  or  circular  arc  w  will  denote  the  logarithms  of  the  numbers 
represented  by  the  corresponding  values  of  r.  Hence,  the  propriety 
of  the  name  logarithmic  spiral.     In  this  curve 

tan.  /  P  =  r  —  — —  =  a    — =  m  ; 

dw  m 

hence  the  curve  cuts  all  its  radii  vectores  under  the  same  angle. 


120  THE  DIFFERENTIAL  CALCULUS. 

9.  To  determine  the  subtangent  at  any  point  in  the  Spiral  of 
Archimedes,  its  equation  being 

r  =  au 

dr  .        dr  „  „ 

•  ••  -r—  =  a  .'.T^  -7-  —r-  =  aijf  =  rw  =  T, 
aw  aw 

so  that  FR  is  equal  to  the  length  of  the  circular  arc  to  radius  r,  com- 
prehended between  FR,  FA ;  when,  therefore,  u  =  ^ir^  the  subtan- 
gent equals  the  length  of  the  whole  circumference.  The  spiral  of 
Archimedes  belongs  to  the  class  of  spirals  represented  by  the  general 
equation 

r  =  aw". 
When  n  =  —  1,  we  have  rw  =  a,  and  the  spiral  represented  is 
called  the  hyperbolic  spiral,  on  account  of  the  analogy  between  this 
equation  and  xy  =  a.     It  is  also  calle^  the  reciprocal  spiral. 

10.  To  determine  the  polar  subtangent  at  any  point  in  the  hy- 
perbolic spiral. 

T,  =a. 

11.  To  determine  the  polar  subtangent  at  any  point  in  the  spiral 

— L 
whose  equation  is  r  =  aw  ^ . 

T  =  2aw^  =  — . 
r 

12.  To  determine  the  polar  subtangent  at  any  point  in  the  para' 

— 
bolic  spiral,  its  equation  being  r  =  aw  2. 

T  =^ 

a"  ' 

13-  To  determine  the  polar  subtangent  at  any  point  in  a  spiral 
whose  equation  is 

(,^  _  ar)  w^  —  1  =  0. 

^'  3    • 

(r'  —  ar)'^ 

Rectilinear  Asymptotes. 

(82.)  A  rectilinear  asymptote  to  a  curve  may  be  regarded  as  a 
tangent  of  which  the  point  of  contact  is  infinitely  distant,  so  that  the 
determination  of  the  asymptote  reduces  to  the  determination  of  the 


TPHE  DIFFERENTIA-L  CALCULUS. 


121 


t&ngent  on  the  hypothesis  that  either  or  both  y'  —  0,  x'  =  0,  the 
portions  of  the  axes  between  the  origin  and  this  tangent  being,  at  the 
same  time,  one  or  both  finite. 

The  equation  of  the  tangent  being 

we  have,  by  making  successively  y  =  0,  x  =  0,  the  foTlovving  ex- 
pressions for  the  parts  of  the  axes  of  x  and  y,  between  the  tangent 
flnd  the  origin,  viz. 

x'  —  ^mdy'—x'-^.     .     .     .(1). 
dii'         ^  dx'  ^  ' 

W 

If  for  a;  =  CO  both  these  are  finite,  thsy  will  determine  two  points, 
one  on  each  axis,  through  which  an  asymptote  passes  If  for  a:  =  cd 
the  first  expression  is  finite  and  the  second  infinite,  the  first  will  de- 
termine a  point  on  the  axis  of  x,  and  the  second  will  show  that  a  line 
through  this  point  and  parallel  to  the  axis  of  y  is  an  asymptote.  If, 
on  the  contrary,  the  second  expression  is  finite,  and  the  first  infinite, 
the  asymptote  will  pass  through  the  point  in  the  axis  of]/,  determined 
by  the  finite  value,  and  will  be  parallel  to  the  axis  o(x. 

When,  however,  asymptotes  parallel  to  the  axes  exist,  they  may 
generally  be  detected  by  merely  inspecting  the  equation,  as  it  is  only 
requisite  to  ascertain  for  what  values  of  a-,  y  becomes  infinite,  or  for 
what  values  of?/,  x  becomes  infinite.  Thus,  in  the  equation  xi/  =  a, 
X  =  0^  renders  j/  =  cjo  »  and  y  =  0  renders  a;  =  co  ,  therefore  the 
two  axes  are  asymptotes.     Again,  in  the  equation 

bx* 

cfiy^  —  ''f  ^  —  ^-p"  =  0,  or  y^  =  — ; 

U'  XT 

it  is  plain  that  x  =  db  a  renders  y  =  oo,  we  infer,  therefore,  that  the 
curve  represented  by  this  equation  has  two  asymptotes,  each  parallel 
to  the  axis  of  y,  and  at  the  distance  a  from  it. 

If  both  expressions  are  infinite,  there  will  be  no  asymptote  corres- 
ponding to  X  =  00  . 
If  both  expressions  are  0,  the  asymptote  will  pass  through  the  origin, 

....  diy' 

and  its  mclmation  d,  to  the  axis  of  x  will  be  determined  by -rr-  ==* 

tan.  6. 

16 


122  THE  DIFFERENTIAL  CALCULUS. 

If  for  J/  =  00  one  or  both  of  the  above  expressions  are  finite,  there 
will  be  an  asymptote,  and  its  position  may  be  determined  as  in  the 
foregoing  cases. 

EXAMPLES. 

(83.)  1.  Let  the  curve  be  the  common  hyperbola,  of  which  the 
equation  is 

b 


a 
dy  _  bx 

"  ^         a  ^/  x'  —  a? 
hence  the  general  expressions  (82)  are 

x^  —  a?  _         a" 

X  X 

and 

6  a?  ba 


X      1 -J 

both  of  which  are  0,  when  a?  =  oo  ;  hence  an  asymptote  passes  through 
the  origin. 
Also 

dw        ,     6  1 

-r-=  ±  —   . 


Ax  a        ^/  a^ 


which  becomes  ±  — ,  when  x  =  oo  ,  therefore,  this  being  the  tangent 
a 

of  the  inclination  of  the  asymptotes  to  the  axis  of  x,  they  are  both  rep- 
resented by  the  equation 


y  =  ±  -  X, 
^  a 


2,  To  prove  that  the  hyperbola  is  the  only  curve  of  the  second 
order  that  has  asymptotes. 

The  general  equation  of  a  line  of  the  second  order,  when  referred 
to  the  principal  diameter  and  tangent  through  the  vertex  as  axes,  is 

if  =  mx  +  nx^, 


THE    DIFFERENTIAL    CALCULUS. 


123 


y 

2tf       _mx  +  2na^  —  2f  _ 
m  +  2«a;             m  +  2nx 

mx 

dy 

m  +  2nx 

dx 

^ 

mx  +  2niP  _  2tf  —  mx  —  2nx'  _ 

mx 

^      ^dx      ^  2ij  2y  2s/wx  +  ju-^ 

Dividing  numerator  and  denominator  of  each  of  these  expressions  by 
jj,  they  reduce  to 

m  ,  ♦»» 
and 


-  +  2»  2  n/^  +  n 

•*  X 

and  these,  when  x  =  oo  ,  or  indeed  when  y  =  cc  ,  become 

m 


and 


2»         2Vn 

Hence  the  curve  will  have  asymptotes,  provided  n  be  neither  0  nor 
negative,  that  is  to  say,  provided  the  curve  be  neither  a  parabola  nor 
an  ellipse,  but  if  it  be  either  of  these,  there  can  exist  no  asymptote  ; 
therefore  the  hyperbola  is  the  only  Une  of  the  second  order  which  has 
asymptotes. 

(84.)  When  the  curve  is  referred  to  polar  coordinates,  then,  since 
the  radius  vector  of  the  point  of  contact  is  infinite  when  the  tangent 
becomes  an  asymptote,  it  follows  that  if  for  r  =  co  the  subtangent 
is  finite,  this  subtangent  may  be  determined  by  (80)  in  terms  of  w, 
and  w  may  be  found  from  the  equation  of  the  curve,  so  that  there 
will  thus  be  determined  a  point  in  the  asymptote  and  its  direction, 
which  is  all  that  is  necessary  to  fix  its  position.  There  will  always 
be  an  asymptote  if  w  is  finite,  for  r  =  oo  .  If,  for  r  =  oo  ,  w  is  al.«o 
00  ,  there  exists  no  asymptote. 

3.  Let  the  curve  be  the  hyperbolic  spiral. 
By  ex.  10,  art.  81 ,  the  subtangent  at  any  point  is  constant,  and  equal 
to  a,  therefore  there  must  be  an  asymptote  ;   also 

a 
by  the  equation  of  the  curve  w  =  -  =  0,  when 

r  =  00  ,  therefore  the  asymptote  is  perpendicular 
to  the  fixed  axis  at  the  distance  a  from  the  pole. 

Neither  the  logarithmic  spiral,  nor  the  spiral  of 
Archimedes  have  an  asymptote. 


124  THE    DIFFERENTIAL    CAtCULCr.- 

■4.  Let  the  spiral  whose  equation  is 


C.2  _  1  1  _  U-' 

be  proposed,  M'hich  admits  of  a  rectilinear  asymptote,  because  w=  I 
renders  r  =  co  .  Ths  direction,  therefore,  of  the  asymptote  is  ascer- 
tained, and  consequently  the  direction  of  the  infinite  radius  vector,- 
sinco  they  must  be  parallel.  It  remains,  therefore,  to  determine  the 
subtangent,  or  distance  of  the  asymptote  from  the  pole 

ctr  _  2au)-^       _         2r^ 

d(ji  (1  —  u~~)^  au^ 

»  _^  rfr  _       ^  2r^   _   "'■^^  _  **  _  rp 
du  ou'  2  2  ' 

because  w  =  1  when  r  =  co  . 

(85.)  Although  we  do  not  propose  to  treat  fully  in  this  place  of 
curvilinear  asymptotes,  yet  we  may  remark  in  passing,  that  if  r  should 
be  finite  a' though  w  be  infinite,  it  will  prove  that  the  spiral  must  be 
continued  for  an  infinite  number  of  revolutions  round  its  pole,  before 
it  can  meet  the  circumference  of  a  circle  whose  radius  is  this  finite 
value.  In  such  a  case,  therefore,  the  •spiral  has  a  circular  asymptote. 
If,  moreover,  the  value  of  r  for  w  =  co  be  greater  than  the  value  of  v 
for  every  other  value  of  w,  the  spiral  will  be  included  within  its  circu- 
lar asymptote,  but,  otherwise,  it  will  be  without  this  circle. 
5.  Thus  in  the  spiral  whose  equation  is 

(r*  —  ar)  cP  —  1  =  0  or  w  =  —        

•«/  »*^  —  ar 

cj  is  infinite  when  r  =  a,  and  for  all  less  values  of  r,  u  is  imaginary  ; 
hence  the  spiral  can  never  approach  so  near  to  the  pole  as  r  =  a,  till 
after  an  infinite  number  of  revolutions,  so  that  the  circumference 
whose  radius  is  a  is  within  the  spiral  and  is  asymptotic. 
If,  on  the  contrary,  the  equation  had  been 

1 


V  ar  —  r* 
then  also  r  =  a  gives  w  =  oo  ,  but  for  all  other  real  values  of  w,  r  is 
less  than  a,  so  that  this  spiral  is  enclosed  by  its  asymptotic  circle,  the 
radius  of  which  is  a. 


4'  Co 


THE  DIFFERENTIAL  CALCULUS. 


125 


6.  To  determine  the  rectilinear  asymptote  to  the  logarithmic 
curve. 

The  axis  of  a?. 

7.  To  determine  the  equation  of  the  asymptote  to  the  curve 
whose  equation  is 

The  equation  isy  =^  x  -\-  \  a. 

8.  To  determine  the  rectilinear  asymptote  to  the  spiral  whose 
equation  is  , 

I 

r  =  ow  - . 

The  fixed  axis  is  the  asymptote. 

9.  To  determine  whether  the  spiral  shown  to  have  a  rectilinear 
asymptote  in  ex.  4  has  also  a  circular  asymptote. 

The  circle  whose  centre  is  the  pole  and  radius  =  a  is  an  asymp- 
tote. 

(86.)  Before  terminating  the  present  chapter,  it  will  be  necessary 
to  exhibit  the  expression  for  the  differential  of  the  arc  of  any  plane 
curve,  as  we  shall  have  occasion  to  employ  this  expression  in  the  next 
chapter. 

Let  us  call  the  arc  AB  of  any  plane  curve  s,  and 
the  coordinates  of  B,  x,  i/ ;  let  also  BD  be  a  tan- 
gent at  B,  and  BC  any  chord,  then  if  BE,  ED  are 
parallel  to  the  rectangular  axes,  BC  will  be  the  in- 
crement of  the  arc  s  corresponding  to  BE  =  h,  the 
increment  of  the  abscissa  x. 

Now,  putting  tan.  DBE  =  a,  we  have 


and 


ED  =  /ia  .-.  BD  =  n/  /i2  +  k'a? 


BD  +  DC  _  ^  h^(^\  +  oC')  -f  /la—  CE 


BC 


x/  A^  +  CE'' 
CE 


This  ratio  continually  approaches  to  t—  or  to  unity  as  h  diminish- 
es and  this  it  actually  becomes  when  h  =  0.  Consequently,  since  the 
arc  BC  is  always,  when  of  any  definite  length,  longer  than  the  chord 


126  THE  DIFPERENTIAL  CALCULUS. 

BC  and  shorter  than  BD  +  BC,*  it  follows  that  when  h  =  0  that 
the  ratio  of  the  arc  to  either  of  these  must  be  unity ;  therefore 

.     ,     ,.    .      arc  BC  arc  BC        chord  BC 

in  the  hmit  -r— 3-^7;  =  1  •*•  — r —   -^ , =  l^ 

chord  BC  h  h 

but 


chord  BC      n/  /i2  +  CE^^  n/j       CE^ 


h  h  h'  ' 

and  CE  is  the  increment  of  the  ordinate  y  corresponding  to  the  incre- 
ment h  of  the*  abscissa  x ;  hence,  when  h  =  0,  the  ratio  becomes 


^  ^  ^1  +  -^  =  1 
dx     '  daP 


ds        V  df 

dx  dx^ 

If  any  other  independent  variable  be  taken  instead  of  x,  then,  denoting 
the  several  differential  coefficients  relatively  to  this  new  variable  by 
((ic),  (dt/),  {ds)  we  have  (66) 


At  the  point  where -^  =  0,—  =  1,  or  (ds)  =  {dx). 


CHAPTER     II. 


ON  OSCULATION,  AND  THE  RADIUS  OF  CURVATURE 
OF  PLANE  CURVES. 

(87.)  Let 

2/=/p.  Y  =  Fx, 

be  the  equations  of  two  plane  curves,  in  the  former  of  which  we  shall 
suppose  the  constants  a,  6,  c,  &c.  to  be  known,  and  therefore  the 
curve  itself  to  be  determinate  ;  while  in  the  latter  we  shall  consider 
the  constants  A,  B,  C,  &c.  to  be  unknown,  or  arbitrary,  and  there- 

*  See  Young's  Elements  of  Plane  Geometry. 


THE  DIFFERENTIAL  CALCULUS.  1'27 

fore  the  species  only  of  the  curve  given.     The  constants  which  enter 
into  the  equation  of  a  curve,  are  usually  called  the  parameters. 

If,  now,  X  take  the  increment  h,  and  the  corresponding  ordinates 
y%  Y'  be  developed,  we  shall  have,  by  Taylor's  theorem, 

Now,  the  parameters  which  enter  (2)  being  arbitrary,  they  maybe 
determined  so  as  to  fulfil  as  many  of  the  conditions 
dy_dY    d^y   _dn 

^^-^'d^-^'d^-d^'^^ ^^^' 

as  there  are  parameters,  but  obviously  not  more  conditions. 

We  shall  thus  have  the  values  of  A,  B,  C,  &c.  in  terms  of  x,  and 
of  the  fixed  parameters  a,  6,  c,  &c. ;  which  values,  substituted  in  (2), 
will  cause  so  many  of  the  leading  terms  in  both  series  to  become 
identical,  whatever  be  the  value  of  a;.  Other  corresponding  terms  of 
the  two  series  may,  indeed,  be  rendered  also  identical,  but  this  can 
take  place  only  accidentally,  not  necessarily.  Hence,  whatever  par- 
ticular value  we  now  give  to  x,  the  resulting  values  of  the  corres- 
ponding coefiicients  will  necessarily  agree  to  the  extent  mentioned, 
that  is,  as  far  as  the  n  first  terms,  if  there  are  n  constants  originally 
in  (2)  ;  and  this  is  true,  even  if  such  particuleir  value  of  x  render  any 
of  the  coefficients  infinite,  inasmuch  as  they  are  always  identical  as 
far  as  these  terms,  but  no  further. 

We  know,  however,  that  in  those  cases  where  any  of  the  coeffi- 
cients become  infinite,  (1)  and  (2)  will  fail  to  represent  the  true  de- 
velopments of  the  ordinates  y',  Y'  at  the  proposed  points.  Neverthe- 
less, as  the  two  series  have  been  rendered  identical,  as  far  as  n  terms, 
should  they  both  fail  within  this  extent,  the  terms  which  supply  these 
in  the  true  development,  must  necessarily  be  identical.  (See  note  C 
at  the  end.)  // 

Now  the  greater,  number  of  leading  terms  in  the  two  developments, 
which  are  identical,  the  nearer  will  the  developments  themselves  ap- 
proach to  identity,  provided,  at  least,  h  may  be  taken  as  small  as  we 
please  ;  for  if  »t  —  1  terms  in  each  are  identical,  we  may  represent 
the  difference  of  the  two  developments  by 


128  THE    DIFFERENTIAL    CALCULUS. 

A„  A'^  +  S  —  {k'X'  +  S') (4), 

where  S,  S'  represent  the  sums  of  the  remaining  terms  in  each  series 

after  the  nth.  Hence,  ^  being  the  highest  power  of  A.  which  enters 
this  expression,  for  the  difference  it  follows  from  (47),  that  a  value 

may  be  given  to  h  small  enough  to  cause  the  term  A„  h  to  become 
greater  than  all  the  other  terms  in  (4),  and  consequently,  for  this 
small  value, 

A„  h"-  —  a;  }/  7  S  —  S', 

and,  therefore,  the  whole  difference  (4)  is  greater  than  twice  S  —  S', 
but  when  the  nth  term  is  the  same  in  both  developments,  as  well  as 
the  preceding  terms,  then  the  difference  (4)  is  reduced  simply  to 
S  —  S',  which  we  have  just  seen  to  be  less  than  (4).  Consequently 
the  developments  approach  nearer  to  identity,  for  all  values  of  h  be- 
tween some  certain  finite  value  h'  and  0  as  the  number  of  identical 
leading  terms  become  greater. 

When  the  first  of  the  conditions  (3)  exist,  the  curves  have  a  com- 
mon point ;  when  the  second  also  exists  they  have  a  common  tangent 
at  that  point,  and  are  consequently  in  contact  there,  and  the  contact 
will  be  the  more  intimate,  or  the  curves  will  be  the  closer  in  the 
vicinity  of  the  point,  as  the  number  of  following  conditions  become 
greater ;  so  that  of  all  curves  of  a  given  species,  that  will  touch  any 
fixed  curve  at  a  proposed  point  with  the  closest  contact  whose  para-^ 
meters  are  all  determined  agreeably  to  the  conditions  (3).  No  other 
curve  of  the  same  species  can,  from  what  is  proved  above,  approach 
so  nearly  to  coincidence  with  the  proposed,  in  the  immediate  vicinity 
of  the  point  of  contact,  as  this  ;  so  that  no  other  of  that  species  can 
pass  between  this  and  the  proposed.  A  curve,  thus  determined,  is 
said  to  be,  in  reference  to  the  proposed  curve,  its  osculating  curve  of 
the  given  species. 

(88.)  It  appears,  from  what  has  now  been  said,  that  there  may  be 
different  orders  of  contact  at  any  proposed  point.  The  two  first  of 
the  conditions  (3)  must  exist  for  there  to  be  contact  at  all ;  therefore, 
when  only  these  exist,  the  contact  is  called  simple  contact,  or  contact 
of  the  first  order ;  if  the  next  condition  also  exist,  the  contact  is  of 
the  second  order,  and  so  on;  and  it  is  obvious,  that  of  any  given, 
species,  the  osculating  curve  will  have  the  highest  order  of  contact^ 


THE  DIFFERENTIAL  CALCULUS.  129 

at  any  proposed  point,  in  a  given  curve.  If  the  curve,  given  in 
species,  has  n  parameters,  the  highest  order  of  contact  will  be  the 
»  —  1th,  unless,  indeed,  the  same  values  of  these  parameters  that 
fulfil  the  n  conditions  (3),  should  happen  also  to  fulfil  the  n  +  1th, 
the  n  +  2th,  &c. ;  but  this,  as  observed  before,  can  take  place  only 
accidentally,  and  cannot  be  predicted  of  any  proposed  point,  although 
we  see  it  is  possible  for  such  points  to  exist. 

(89.)  At  those  points  in  the  proposed  curve,  for  which  Taylor's 
development  does  not  fail,  contact  of  an  even  order  is  both  contact 
and  intersection,  and  contact  of  an  odd  order  is  without  intersection ; 
before  proving  this,  however,  we  may  hint  to  the  student  that  contact 
is  not  opposed  to  intersection,  for  two  curves  are  said  to  be  in  con- 
tact at  a  point,  when  they  have  a  common  tangent  at  that  point ;  and 
yet,  as  we  are  about  to  show,  one  of  these  curves  may  pass  between 
the  tangent  and  the  other,  and  so  intersect  where  they  are  admitted  to 
be  in  contact.  To  prove  the  proposition,  let  us  take  the  difference 
(4),  which,  when  Taylor's  theorem  holds,  is 

(A„  —  A'„)  A""  +  S  —  S' (5,) 

A„  A'„  being  here  the  n  —  1th  differential  coefficients.     If  these  are 

odd,  the  contact  is  of  an  even  order,  also  a  being  odd,  h""  will  have 
contrary  signs  for  h  =  -\-  h'  and  h  =  —  ft',  and  therefore,  since  for 
these  small  values  of  ft,  the  sign  of  the  whole  expression  (5)  is  the 
same  as  that  of  the  first  term,  the  differences  of 
the  ordinates  corresponding  to  a?  +  ft,  and  to 
X  —  ft,  will  be  the  one  positive  and  the  other 
negative,  so  that  the  two  curves  must  necessarily 
cross  at  the  point  whose  abscissa  is  x. 

But  if  a  is  even,  the  contact  is  of  an  odd  order,  and  the  difference 
(5)  between  the  ordinates  of  the  two  curves  corresponding  to  the 
same  abscissa,  a:  +  ft,  will,  for  a  small  value  of  ft,  have  the  same 
sign,  whether  ft  be  positive  or  negative ;  so  that,  in  this  case,  the 
curves  do  not  cross  each  other  at  the  point  of  contact. 

(90.)  The  student  must  not  fail  to  bear  in  remembrance,  that  the 
proposition  just  established,  comprehends  only  those  points  of  the 
proposed  curve,  at  which  none  of  the  differential  coefficients  become 
infinite  from  the  first  to  that  immediately  beyond  the  coefficient  which 
fixes  the  order  of  the  contact.     For  it  is  only  upon  the  suppositicm 


130  THE  DIFFERENTIAL  CALCULUS. 

that  the  true  development,  within  the  limits,  proceeds  according  to 
the  ascending  integral  and  positive  powers  of  h,  that  the  foregoing 
conclusions  respecting  the  signs  of  the  difference  (5)  can  be  fairly 
drawn.     (See  note  C.) 

(91.)  From  the  principles  of  osculation  now  established,  it  is  evi- 
dent that  any  plane  curve  being  given,  and  any  point  in  it  chosen,  we 
may  always  find  what  particular  curve,  of  any  proposed  species,  shall 
touch  at  that  point  with  the  closest  contact,  or  which  shall  most 
nearly  coincide  with  the  given  curve  in  the  immediate  vicinity  of  the 
proposed  point.  Thus  an  ellipse  or  a  parabola  being  given,  and  a 
point  in  it  proposed,  we  may  determine  the  circle  that  shall  approach 
more  nearly  to  coincidence  with  that  ellipse  or  parabola  in  the  vicinity 
of  the  proposed  point,  than  any  other  circle,  and  which  will  therefore 
better  represent  the  curvature  of  the  given  curve  at  the  proposed  point 
than  any  other.  On  account  of  its  simpUcity  and  uniformity,  the 
circle  is  the  curve  employed  to  estimate,  in  this  way,  the  curvature  of 
other  curves  at  proposed  points;  that  is,  the  curvature  is  estimated  by 
the  curvature  of  the  osculating  circle,  or  rather  as  the  curvature  of 
a  circle  increases  as  the  radius  diminishes,  and  vice  versa,  it  is  usual 
to  adopt,  as  a  fit  representation  of  the  curvature,  the  reciprocal  of  the 
radius. 

The  osculating  circle  is  called  the  circle  of  curvature,  and  its  ra- 
dius the  radius  of  cimature,  and,  from  what  has  been  said  above,  it 
follows  that  the  determination  of  the  curvature  at  any  point  in  a  pro- 
posed curve,  reduces  itself  to  the  determination  of  this  radius :  to  this, 
therefore,  we  shall  now  proceed. 

Radius  of  Curvature. 

FROBLEM  I. 

(92.)  To  determine  the  radius  of  curvature  at  any  proposed  point 
of  a  given  curve. 

The  general  equation  of  a  circle  being 

(^x~uY+{y—l3y  =  r\ 
it  becomes  determined  as  soon  as  we  fix  the  values^f  the  parameters 
a,  jS,  r,  and  these  may  be  determined,  so  as  to  fiilfil  any  three  inde- 
pendent conditions,  but  not  more.     In  the  case  before  us,  the  condi- 
tions to  be  fulfilled  are  those  of  (3)  art.  (87),  that  is  to  say,  putting 


THE    DIFFERENTIAL    CALCULUS.  131 

p',  p",  &c.  for  the  successive  differential  coefficients  derived  from  Y 
=  Fx,  the  equation  of  the  given  curve,  the  conditions  to  be  fulfilled 
are 

y   ^Ux  ^'dx"   ^' 

in  order  that  the  resulting  values  of  a,  ^,  r,  may  belong  to  the  equa- 
tion of  the  osculating  circle.     Now 

dy  _       X  —  a  dy-  _  1  (^  —  o-Y r^  , 

di       f^ij^'t?       ^^^     W—^f"     iy  —  ^r 

hence  the  three  equations  for  determining  a,  (3  and  r,  are 
(X  — a)+l>'(2/  — /3)=0..,.(2), 
From  the  second  equation 

{x-o,r  =  p'^y-^r. 

Substituting  this  in  the  first, 

ip'  +  1)  (y  ~  I^T  =  r"' 

Adding  this  last  to  the  third,  there  results 


y-^  =  -'-^' 

? 

which, 

substituted  in  (2),  gives 

X  —  a  — 

p"     ■ 

Consequently, 

a  =  x 

.  p"  + 1 

y+  p" 

da^' 

r : 

P"- 

These  equations  completely  determine  the  osculating  circle,  when- 
ever the  co-ordinates  x,  y  of  the  proposed  point  £ire  given. 

Should  this  point  be  such  as  to  render  p  =  0,  then  the  expression, 
for  the  radius  of  curvature  at  that  point,  becomes 


182  THK  DIFPERENTIAL  CALCULTTS. 

p"      w 

dor' 

But  when  p'  =  0,  the  tangent  at  the  proposed  point  must  be  pa- 
rallel to  the  axis  of  x  (78),  or,  which  is  the  same  thing,  the  axis  of  y 
must  coincide  with  the  normal ;  hence,  under  this  arrangement  of  the 
axes,  x  =  0  at  the  proposed  point,  and  therefore 

Should  p"  =  0  at  the  proposed  point,  r  will  be  infinite,  whether 
jj'  =  0  or  not,  so  that  the  osculating  circle  then  becomes  a  straight 
line ;  as,  therefore,  this  straight  line  has  contact  of  the  second  order, 
/  the  parts  of  the  curve  in  the  vicinity  of  the  point 
"y^  will  lie  on  contrary  sides  of  it,  as  in  the  annexed 

diagram  (89),  that  is,  supposing  p'"  is  neither  0  nor 
CO  .  Ifp'"  =  0,  and  the  next  following  differential  coefficient  nei- 
ther 0  nor  C30  ,  the  contact  will  be  of  an  order  which  is  unaccompanied 
by  intersection. 

A  point  at  which  the  tangent  intersects  the  curve,  or  at  which  the 
curve  changes  from  convex  to  concave,  is  called  a  point  of  inflexion^ 
or,  a  point  of  contrary  flexure.  The  analytical  indications  of  such 
points  will  be  more  fully  inquired  into,  when  we  come  to  speak  of  the 
singular  points  of  curves. 

(93.)  By  referring  to  equation  (2)  above,  which  has  place  even 
when  the  contact  is  but  of  the  first  order,  we  learn  that  the  centre 
(a,  |8)  of  every  touching  circle,  is  always  on  the  normal  at  the  point 
of  contact ;  for  that  equation  is  the  same  as 

dx 
We  shall  now  apply  the  general  expression,  for  the  radius  of  cur- 
vature, to  a  few  particular  cases. 

EXAMPLES. 

(94.)  1.  To  determine  the  radius  of  curvature,  at  any  point  in  a 
parabola. 


THE  DIFFERENTIAL  CAIiCULUg.  133^ 


Differentiating  the  equation  of  the  curve, 
y^  =  4mx, 
we  have, 

2yp  =  4m  .•.  p  =  — 
2yp"-{-2p-  =  0.'.p"  =  ^  =  -^ 

= r —  (See  Anal.  Geom.) 

4nr 

As  the  expression  for  the  normal  dimini^es  with  x,  the  vertex  is 
the  point  of  greatest  curvature,  r  being  there  equal  to  2m,  or  to  half 
the  parameter. 

2.  To  determine  the  radius  of  curvature  at  any  point  in  an  el- 
lipse. 

By  differentiating  the  equation 

ay  +  6V  =  oFl^, 
we  have 

a)fp'  +  b^x  =  0  .'.  p'  = 


ahjp"  +  ay2_j.  5a  =  o.-.p 


„  _       6'  +  ay 


a^y  ay 


''•*"  p"  ay          *   6^  a'b*         "  ^^' 

From  this  expression,  others  occasionally  useful  may  be  readily 

derived.     Thus,  since  {Anal.  Geom.)  the  square  of  the  normal,  N,  is 

b* 

— -  ar"  +  «^,  therefore. 


« 


a*N^  =  6V  +  aV .-.  r  =  ^  =  -^  N^  .  .  .  (2). 
Again,  since  {Anal.  Geom.), 

aN  =  66' .-.  r  =  ^  .  .  .  .  (3). 


134  THE  DIFFERENTIAL  CALCULUS. 

At  the  vertex  r  =  —  =  semiparameter  {Anal.  Geom.) 

From  equations  (2)  and  (3)  it  follows  that,  in  the  ellipse,  the  radius 
of  curvature  varies  as  the  cube  of  the  normal,  or  as  the  cube  of  the 
diameter  parallel  to  the  tangent  through  the  proposed  point. 

It  is  often  desirable  to  obtain  *•  as  a  function  of  X,  the  angle  inclu- 
ded between  the  normal  and  the  transverse  axis.  For  this  purpose 
we  have  since 

ar^  =  o^  (1  _  |!.)  and  f  =  TS^  sin.=^ 

a^  b^      ' 

.•.NM1-(1— ^)sin.^|=-^ 


but  {Anal.  Geom.) 


.■.N=-^ I 

a 


(1  —  e^  sin.2X)2 
=  ^  N3  ^  ^1^ a  (1  —  e^) 


a  (1  —  e^  sin.  ^X)  ^      ( 1  —  e^  sin.  ^X)  ^ 

(95.)  Since,  in  the  ellipse,  the  principal  transverse  is  the  longest 
diameter,  and  its  conjugate  the  shortest  {Anal.   Geom.),  it  follows 

from  (3),  that  the  curvature  -  is  greatest  at  the  vertex  of  the  trans- 
verse, and  least  at  the  vertex  of  the  conjugate  axis.     At  the  former 

fe2  a^ 

point  r  =  -T ,  and  at  the  latter  r  =  -j-. 

The  present  is  a  very  important  problem,  being  intimately  con- 
nected with  inquiries  relative  to  the  figure  of  the  earth. 

By  means  of  the  last  expression  for  r,  the  ratio  of  the  polar  and 
equatorial  diameters  of  the  earth,  may  be  readily  deduced,  when  we 
know  the  lengths  of  a  degree  of  the  meridian  in  two  known  latitudes, 
L,  I,  for  these  lengths  may,  without  error,  be  considered  to  coincide 
with  the  osculating  circles  through  their  middle  points ;  and  since 


THE    DIFFERB9rTIA.L  CALCULUS.  135 

similar  arcs  of  circles  are  as  their  radii,  we  have,  by  putting  M,  m  for 

the  measured  degrees,  and  R,  r  for  the  corresponding  radii, 

R  :  r  :  :  M  :  m, 

but 

g  (1  -  e')                  _       «(l-e^) 
R  =   ^^ '. — -  and  r  = -^ 

(1  — e2sin.2L)2  (1  — e==sin.2/)2 

therefore,  since  r»R  =  Mr,  we  have 

m  M 


(1— e^sin.^L)^       (1— e^sin.^/)^ 


or 


m\(l  —  <?  sm.^l)  =  M3  (1  —  e^sin-^'L), 

62  M^  — m3 

.'.e^=  1 


"""       M^sin-^L  — m3sin.2i 


o_    ,  .  M3  sin.  ^L— m3sin.2i 

•••  1  -  V  I  -_ ^ 

•  m^cos.^/  — M^cos.^'L 

2. 

sin.'^L  — (^sin.^/ 

=  ^{^ }• 

(:jr|)    COS.^/ COS.^L 

If  /  =  0,  that  is,  if  the  degree  m  is  measured  at  the  equator,  then, 
a  ,  sin.  L  , 

3.  To  determine  the  radius  of  curvature  at  any  point  in  the  loga- 
rithmic curve,  its  equation  being  y  =  a', 

(m*  +  f)^       ,    .        . 

r  = :i-^  ,  m  beme  the  modulus. 

my 

4.  To  determine  the  radius  of  curvature  at  any  point  in  the  cu- 
bical parabola,  its  equation  being  ■f  =  ax. 

Qa'y 


136  THE  DIFFERENTIAL  CALCULUS. 

PROBLEM    II. 

(96.)  To  determine  these  points  in  a  given  curve,  at  which  the 
osculating  circle  shall  have  contact  of  the  third  order. 

It  is  here  required  to  find  for  what  points  of  a  given  curve  the  val- 
ues of  a,  (3,  r,  determined  by  the  three  first  conditions  (3),  art.  (87) 
satisfy  also  the  fourth  condition. 

The  differential  coefficient  p'"  as  derived  fi"om  equation  (3),  p. 
131,  is 

and  this  must  agree  with  the  p"  derived  from  the  equation  of  the  pro- 
posed curve,  at  those  points  where  the  contact  is  of  the  third  order  ; 
that  is,  the  abscissas  of  these  points  will  all  be  given  by  the  roots  of 
the  equation 

{y  —  (3)p"'  +  Sp"p'  =  o, 

and  it  may  be  easily  shown,  that  the  points  which  satisfy  this  equa- 
tion are  those  of  greatest  and  least  curvature,  for  since 

,_        (P"+1)^ 
P" 

.   ^^  _  —  3  ip"  +  1)^  p'p'""  +  ip'^  +  l)^p" 
**•  di  ~  p"^ 

and  when  r  is  a  maximum  or  a  minimum  this  expression  is  equal  ta 
0  (49) ;  hence 

—  3p'p"^-\.{p'^'+  l)p"'  =  0, 

or,  dividing  by  p"  and  recalling  the  value  of  y  —  ^  deduced  in  (92),, 
we  have,  finally, 

{y^^)p"'+3p"p'  =  0, 

which  being  the  same  equation  as  that  deduced  above,  it  follows  that 
the  points  of  maximum  and  minimum  curvature  are  the  same  as  those 
at  which  the  contact  is  of  the  third  order. 

(97.)  In  the  preceding  investigations  we  have  always  considered 
X  to  be  the  independent  variable,  because  the  expression  for  the  ra- 
dius of  curvature  has  been  obtained  conformably  to  this  hypothesis. 
But  if  any  other  quantity  is  taken  for  the  independent  variable,  the 


THE  DIFFERENTIAL  CALCULUS,  137 

foregoing  expression  for  r  will  not  apply  ;  therefore,  in  order  to  give 
the  greatest  generality  possible  to  the  formula  for  the  radius  of  cur- 
vature, we  shall  now  suppose  any  arbitrary  quantity  whatever  to  be 
the  independent  variable,  x  and  y  being  functions  of  it.  Hence,  in- 
stead ofp'  and  p",  we  shall  have  (66) 

&)  and  {d'y){dx)-{d'x){dy) 

(dx)  {dx') 

the  parentheses  intending  to  intimate  that  the  independent  variable, 
according  to  which  the  differentials  of  the  functions  x,  y  are  taken,  is 
arbitrary,  and  the  differential  of  which  when  chosen  is  with  its  proper 
powers  to  be  introduced  as  denominators  of  the  above  differentials. 
Making,  therefore,  these  substitutions  in  the  expression  for  r,  it  be- 
comes 

( {dyr  +  {dxf  )i 


{dFy)  (dx)  -  {dJ^x)  {dy) 
■or,  since  (86) 


{ds)=^{dyr-\-idxr, 

whatever  be  the  independent  variable, 

(dsy 


.  (1). 


{d^y)  {dx)  -  {d'x)  {dy) 

(98.)  This  expression  is  of  the  utmost  generality,  and  will  furnish 
a  correct  formula  for  every  hypothesis  respecting  the  independent  va- 
riable.    Thus,  if  X  be  chosen  for  the  independent  variable,  then 
(dx)  =  1  and  (d^x)  =  0,  and  the  formula  in  that  case  is 
d£_ 

~d^ 
r-^  ....  (2). 

dx" 
being  the  same  as  that  at  first  given  as  it  ought  to  be.     Ify  be  the 
independent  variable,  then  (dy)  =  1  and  (di'y)  =  0,  so  that  upon  this 
hypothesis  the  formula  is 

.  =  ^^....(3). 

dy' 

18 


138  THE  DIFFERENTIAL  CALCULUS. 

If  5  be  the  independent  variable,  then  {ds)  =  1 ; 

.-.  id^s)  =  d  VJWWyr  -  0  .-.  {d^y)  =-^^{d^x)  .  . .  (4). 

substituting  this  in  the  denominator  of  (1)  we  have 

dy 

.  -  _  »)  ^  (U^^-JJff  =_  (M  =_5  .  .   .  .  (5). 


(d'x)  ■   '     '         '  ^'  (d^x)  _d^ 

By  squaring  ( 1 )  on  this  last  hypothesis  we  have 

r'=  ' 

but,  since  from  (4) 

(d'y)  {dy)  +  {d'x)  {dx)  =  0, 
it  may  be  added  to  the  denominator  of  this  expression  for  r^  without 
affecting  its  value,  so  that 

1 


f^  = 


{d^y)  {dx)  -  {d^x)  {dy)  Y-\-{dhj)  {dy)  +  {d^x)  {dx)  Y 
1 


(dy)2+  (rfx)2  X  {dY/-\-{dPx)'' 
1  1 


(d^y)^  +    {di'x)^  1      rf2„  dX. 


(6). 


(99.)  We  shall  now  proceed  to  determine  a  suitable  formula  for 
polar  curves. 

If  the  circle  whose  equation  is  (1)  p.  131,  be  transformed  from 
rectangular  to  polar  coordinates,  the  pole  being  at  the  origin  of  the 
primitive  axes,  and  the  axis  of  x  being  the  fixed  line  from  which  the 
variable  angle  u  of  the  radius  vector  y  is  measured,  we  shall  have 
{Anal.  Geom.) 

(y  COS.  Gj  —  a)^  +  {y  sin.  w  —  (3^)  =  r^  .  .  .   .  (1). 
If,  therefore,  we  differentiate  on  the  supposition  that  w  is  the  inde- 
pendent and  y  the  dependent  variable,  and  denote  the  first  and  second 
differential  coefficients  by  p,  and  p,,,  we  shall  have 


THE  DIFFERENTIAL  CALCULUS.  139 

(ycos.a)  — a)  (;?,cos.&)— y sin. to)  +  (y  sin.  <o—/i)  (;), sin. to  +  y cos.  w)  =  0 . . .  (2) 
(j9,cos.o>  —  y  sin.  &))''+ (y  COS.  w  — a)  (p„  COS.  w — 2/?,sin.ft»  —  ycos.w)  + 
{p, sin. 0)  +  y  COS.  &))2-j-  (y  sin.  to — 0)  (p„ sin. a  +  2/), cos. lo — y  sin.  to)  =  0 ...  (3) 
If  from  the  two  latter  equations  we  determine  the  values  of  y  sin. 
w  —  (3  and  y  cos.  w  —  a,  and  substitute  them  in  (1),  we  shall  obtain 
the  following  expression  for  r  in  functions  of  y  and  its  differential 
coefficients,  viz. 

_Jf  +  p^    ....    (4) 

■f  +  ^f'-m, 

But  we  shall  arrive  at  this  expression  more  readily  by  first  deducing 
from  the  equations  «• 

y  =  Y  sin.  u,  x  =  y  cos.  w 
the  differential  coefficients 

—^  =  y  COS.  u  -{-p,  sin.  w  =  (dy) 
-T—  =  —  y  sin.  w  +  p^  cos.  w  =  (da?) 

-T-^  =  —  y  sin.  w  +  2p^  cos.  w  +  p^^  sin,  w  =  (cFi/) 

cPx 

-5-5-  =  —  y  cos.  w  —  2p^  sin.  u  +  p^^  cos.  w  =  (d^a?) 

and  then  substituting  them  in  the  general  formula  (1). 
Since  (80)  the  expression  for  the  normal  PN  is 

N  -  y^  +  p^K 
we  may  put  the  above  expression  for  r  under  the  form 
_  W 

''""^M^P?  — y/?,    ■  *  ■  ■  ^^^' 
5.  To  determine  the  radius  of  curvature  at  any  point  in  the  loga- 
rithmic spiral 


dry   a       y    

du         m         m 
dPy   _    y 
'd^~~^  ~  ^"' 


140  THE  DIFPBRKNTIAL  CALCULUS. 

Hence 


(y^    _1_    „  2^2  1  I  1 

I  I 

y  v/ 1  -I- (art.  80)  =  y  cosec  P. 

tan.-  P 

It  appears,  therefore,  that  the  radius  of  curvature  is  always  equal  to 

the  normal. 

6.  To  determine  the  radius  of  curvature  at  any  point  in  the  curve 

whose  equation  is 

y  =  2  cos.  w  ±  1 

3 

(5  ±  4  cos.  u)- 

.*.  r  = • 

9  ±  6  COS.  cj 


CHAPTER  III. 


ON  INVOLUTES,  EVOLUTES,  AND  CONSECUTIVE 
CURVES. 

(100.)  If  osculating  circles  be  applied  to  cwr?/ point  in  a  curve, 
the  locus  of  their  centres  is  called  the  evoliUe  of  the  proposed  curve, 
this  latter  being  called  the  involute. 

The  equation  of  the  evolute  may  be  determined  by  combining 
the  equation  of  the  proposed  curve  with  the  equations  (2),  (3)  p.  131, 
containing  the  variable  coordinates  a,  /3  of  the  centre.  As  these 
three  equations  must  exist  simultaneously  for  every  point  of  contact 
[x,  y),  the  two  quantities  x,  xj  may  be  eliminated,  and  therefore,  a 
resulting  equation  obtained  containing  only  a  and  j8,  which  equation 
therefore  will  express  the  general  relation  between  a  and  jS  for  every 
point  {x,  y) ;  in  other  words,  it  will  represent  the  locus  of  the  centres 
of  the  osculating  circles. 

Or,  representing  the  equation  of  the  proposed  curve  by  t^  =  Fx, 
we  shall  have  to  eliminate  x  and  y  from  the  equations  (p.  131) 


THE  DIFFERENTIAL  CALCULUS. 


141 


y  =  F^» 

when  the  resulting  equation  in  a,  /3  will  be  that  of  the  evolute. 


EXAMPLES. 

(101.)  To  determine  the  evolute  of  the  common  parabola 
f  =  4mx.'.p'  =  —  .'.p"  =  —  — 

.'.  1  -{-  p^  -  y^  +  ^"^'  =  1  +  ^,^  =  _i 

*  *  '^  y^  x^  p''  2m 

.     y^     ,  «      ,    «  a  —  2m 

.*.  a  =  ar  +  ■—-  +  2m  =  3a;  +  2m  .-.  x  = 


^  =  y 


2m 


7/=*  _—f_  —  2:r^    ^       _    wijS^  ^ 

Am^~y  y^  T~  '''  ^  ~    ~^ 

...   -f  111,2 


.•./3^ 


27 


(a  —  2m)^ 


which  is  the  equation  of  the  evolute.  If  the  origin  be  removed  to 
that  point  in  the  axis  of  x  whose  abscissa  is  2m,  then  the  equation  be- 
comes 

The  locus  of  which  is  called  the  semicubical  parabola. 
It  passes  through  the  origin  because  ^  =  0  when 
a  =  0  ;  therefore  the  focus  of  the  proposed  involute  is 
in  the  middle,  between  its  vertex  and  the  vertex  of  the 
evolute.  {Anal.  Geom.  art.  100.)  The  curve  con- 
sists of  two  branches  symmetrically  situated  with  respect  to  the  axis 
of  a?  or  of  a,  and  lies  entirely  to  the  right  of  the  origin,  for  every  posi- 
tive value  of  a  gives  two  equal  and  opposite  values  of  (3,  and  for 
negative  values  of  a,  (3  is  impossible.  It  is  easy  to  see,  there- 
fore, that  the  form  of  the  curve  is  that  represented  in  the  margin. 
2.  To  determine  the  evolute  of  the  ellipse. 
By  example  2,  page  133,  we  have 


142  THE  DIFFERENTIAL  CALCULUS. 

b'x       „  b* 

P     — 2-'  P       = —3 

Now,  since,  by  the  equation  of  the  curve, 

.'.  a'f  +  b'x'  =  6"  {a'  —  c'x')  or  =  a"  {¥  +  c^)' 
c^  being  put  for  a^  —  6^.     Hence,  by  substitution, 

Substituting  these  values  in  the  equation  of  the  involute,  we  have 

c  c 

a^b^ 
or,  finally,  dividing  all  the  terms  by  — ,  we  obtain  for  the  evolute  the 


A' 
C3 


equation 


If  a  =  0,  then  (3  =  ±  -j-,  so  that  the  curve  meets 
6 

the  axis  of  y  in  two  points,  c,  rf,  equidistant  from 

the  origin  0.     If /3  =  0,  then  a  =  ±  — ,  so  that 

a 

it  also  meets  the  axis  of  a;  in  two  points,  6,  a,  equi- 
distant  from  0.     If  a  is  numerically  greater  than  — theordinatesbe- 

come  imaginary,  and  if  ^  is  numerically  greater  than  -r-the  abscissa 

becomes  imaginary ;  therefore  the  curve  is  limited  by  the  four  points 
a,  6,  c,  d,  and  touches  the  axes  at  those  points.  It  consists,  there- 
fore, of  four  breinches  symmetrically  situated  as  in  the  figure. 


THE  DIFFERENTIAL  CALCULUS.  143 

3.  To  detemiine  the  evolute  of  the  rectangular  hyperbola,  its 
equation  between  the  asymptotes  being  ocy  =  a^. 
The  equation  of  the  evolute  is 

2 
2.  2  flS 

(a  +  /3)3_(a_/3)^  =-. 
43 

THEOREM. 

(102.)  Normals  to  the  curve  are  tangents  to  the  evolute. 

Let  the  equations  of  the  curve  and  of  its  evolute  be 
y  =  Fx  and  (3  =  fa, 
then  differentiating  the  equation  (2)  p.  131,  considering  a,  ^  as  va- 
riables as  well  as  x,  y,  we  have 

,_|  +  p..(,_«  +  ,-=_p.f  =  0, 

but  (130) 

Hence,  by  substitution, 

^+p'jf  =  0 

ax  ax 

dB       da      d^  1        0  —  y, 

.'. -f- —  ^  or -J- = -  =  ' ^  fequa.2,p.  131). 

dx,        dx       da  p         a  — x  -^         ' 

dB 
Now  -T—  expresses  the  trigonometrical  tangent  of  the  angle  be- 
tween the  axis  of  x  and  a  linear  tangent  through  any  point  (a,  j8)  of 
the  evolute,  and ;-  expresses  the  trigonometrical  tangent  of  the 

angle  between  the  axis  of  x  and  a  normal  at  any  point  {x,  y)  of  the 
involute ;  but  this  normal  necessarily  passes  through  a  point  (a,  (3) 
of  the  evolute,  and,  therefore,  in  consequence  of  the  above  equality, 
it  must  coincide  with  thq  tangent  at  that  point. 

THEOREM. 

(103.)  The  difference  of  any  two  radii  of  curvature  is  equal  to  the 
arc  of  the  evolute  comprehended  between  them. 
Differentiating  the  equation 


THE  DIFFERENTIAL  CALCULUS. 


on  the  hjrpothesis  that  a  is  the  independent  variable,  we  have 


but  by  last  article 


y-f3  =  ia:-a)^ 


and 


_(._„K^ +„=.*...(.). 


Dividing  (2)  by  the  square  root  of  (1)  we  have 
that  is  (86) 


da?  da 


ds        dr  ■ 

—  =-j-  .'.  —  s  =r  ±  a  constant, 

da,        da. 

for  otherwise  there  could  not  be —  =  -r-. 

da        da 

Hence  if  r,  v  be  the  radii  of  curvature  of  any  two  points,  and  s,  s' 

the  corresponding  arcs  of  the  evolute,  then 

r  ±  const.  =  —  s 

r'  ±  const.  =  —  s' 


s. 


so  that  the  difference  of  the  two  radii  is  equal  to  the  arc  of  the  evolute 
comprehended  between  them ;  therefore,  if  a  string  fastened  to  one 
extremity  of  this  arc  be  wrapped  round  it  and  continued  in  the  direc- 
tion of  the  tangent  at  the  other  extremity  as  far  as  the  involute  curve, 
the  portion  of  the  string  thus  coinciding  with  the  tangent  will  by  (102) 
be  the  radius  of  curvature  at  that  point  P  of  the  involute  curve  which 
it  meets,  and,  consequently,  by  the  above  property,  if  the  string  be 
now  unwound,  P  will  trace  out  the  involute. 


THE  DIFPERKNTIAL  CALCULUS.  146 

On  Consecutive  Lines  and  Curves. 

(104.)  Every  equation  between  two  variables  aiay  always  be  con- 
sidered as  the  analytical  representation  of  some  plane  curve,  given  in 
species  by  the  degree  of  the  equation,  and  determinable  both  in  form 
and  position  by  the  constants  which  enter  it,  provided,  these 
constants  are  fixed  and  determinate.  If,  however,  the  equation  con- 
tains an  arbitrary  or  indeterminate  constant  a,  then,  by  assuming  dif- 
ferent values  for  a  the  equation  will  represent  so  many  different  curves 
varying  in  form  and  position,  but  all  belonging  to  the  same  family  of 
curves. 

Now  if  we  consider  the  form  and  position  of  one  of  these  curves  to 
be  fixed  by  the  condition  a  =  a',  another,  intersecting  this  in  some 
point  (x',  y'),  may  be  determined  from  a  new  condition  a  =  a'  +  ^ ; 
and  if  ^  be  continually  diminished,  this  latter  curve  will  approach  more 
and  more  closely  to  the  fixed  curve,  and  will  at  length  coincide  with 
it.  During  this  approach,  the  point  of  intersection  {x,  y')  necessarily 
varies,  and  becomes  fixed  in  position  only  when  the  varying  curve 
becomes  coincident  with  the  fixed  curve.  In  this  position  the  point 
is  said  to  be  the  intersection  of  consecutive  curves,  so  that  what  mathe- 
maticians call  consecutive  curves,  are,  in  reality,  coincident  curves, 
and  the  point  which  has  been  denominated  their  point  of  intersection 
may  be  determined  as  follows : 

(105.)  Let 

F{x,y,x')  =  0 (1) 

represent  any  plane  curve,  x  being  a  parameter,  and  for  any  inter- 
secting curve  of  the  same  family  let  x'  become  x'  +  h,  then,  since 
however  numerous  these  intersecting  curves  may  be,  the  x,  y  of  the 
intersections  belong  also  to  the  equation  (1)  ;  it  follows  that  as  far  as 
these  points  are  concerned,  the  only  quantity  in  equation  (1)  which 
varies  is  x',  therefore,  considering  x,  y  as  constants  in  reference  to 
these  points,  we  have,  by  Taylor's  theorem, 

F(x,j/,x'-i-;i)=F(:r,i/,a:')+    ^^^ /^ + 
'PF{x,y,  x')       h?   ■■        . 

—17^ —  r:^  +  ^^- 

but  F  (x,  y,  x)  =  0,  therefore 

19 


146  THE  DIFFERENTIAL  CALCULUS. 

F  (.r,  y,  X'  +  h)  __  (IF  (t,  y,  x')  drF  (y,  y,  x')      h  ^^ 

h  ~  dx'  dx'^  1-2 

hence,  when  the  curves  are  consecutive,  that  is  when  /t  =  0,  we  have 
the  following  conditions,  viz. 

F  {x,  y,  x')  =  0  \ 
dF  (^,  y,  X')  ^  ^  J  ....  (2) 
dx  * 

to  determine  x  and  y. 

Suppose,  for  example,  it  were  required  to  determine  the  point  of 
intersection  of  consecutive  normals  in  any  plane  curve. 

Representing  the  equation  of  the  curve  by 

y'  =  Fx\ 

and  any  point  in  the  normal  by  (ar ,  y),  we  have  for  the  equation  of  the 
normal 

y  —  y'  =  —  -r{^—x')oT  {y  —  y')p'  +  x  —  x'  =  0. 

This  corresponds  to  the  first  of  equations  (2),  x'  being  the  parameter ; 
hence,  differentiating  with  respect  to  x  of  which  y'  is  a  function  given 
by  the  equation  of  the  curve,  we  have 

{y-y')p"-p"--^=o 

p"  +  1 


•••  !/  =  !/'  + 


P" 


P" 
hence  (92)  consecutive  normals  intersect  at  the  centre  of  curvature. 
(106.)  If  we  eliminate  the  variable  parameter  x' by  means  of  the 
equations  (2),  the  resulting  equation  will  belong  to  every  point  of  m- 
tersection  given  by  every  curve  of  the  family 

F{x,y,x,x')  =0  .  .  .  .  (1), 

and  its  consecutive  curve ;  for  whatever  value  we  suppose  x'  to  take 
in  the  equations  (2),  the  result  of  the  elimination  will  obviously  be 
always  the  same.  Hence  this  resulting  equation  represents  the  locus 
of  all  the  intersections,  and  we  may  show  that  at  these  same  inter- 
sections this  locus  touches  every  individual  curve  in  the  family.  The 
equation  (1),  where  x' represents  a  function  of  x,  i/,  determined  by 


THE  DIFFERENTIAL  CALCULUS. 


147 


the  second  of  the  conditions  (2)  in  last  article,  is  obviously  the  equa- 
tion of  the  locus  of  which  we  are  speaking,  and  the  same  equation, 
when  «' takes  all  possible  values  from  0  to  ±  go,  furnishes  the  family 
of  curves,  which  we  are  now  to  show  are  all  touched  by  this  locus. 
Taking  any  one  of  this  family,  and  differentiating  its  equation  (1),  x' 
being  constant,  we  have 

,         du  .     ,   du  . 
du  =  -T-  dx  +  -T-  ay  =  0. 
dx  dij 

Differentiating  also  the  equation  (1)  of  the  locus,  x'  being  given  by 

the  second  condition  of  (2)  in  last  article,  we  have 

du        ,du        ,    du         _ 

but  by  the  condition  just  referred  to  -j-^  =  0  at  the  point  where  the 

curves  whose  equations  we  have  just  differentiated  meet;  hence, 
since  at  those  points  each  of  these  equations  give  the  same  value  for 

-^,  it  follows  that  they  have  contact  of  the  first  order;  we  infer, 
dx 

therefore,  that  the  equation  ( 1 ) ,  when  x'  is  determined  from  the  second 

of  the  conditions  (2)  last  article,  represents  a  curve  which  touches  and 

envelopes  the  entire  family  of  curves  represented  by  equation  (1),  x' 

being  any  arbitrary  constant.     Thus,  as  we  already  know,  the  locus 

of  the  intersections  of  normals  with  their  consecutive  normals  is  a 

curve  which  touches  them  all  at  their  points  of  intersection,  being  the 

evolute  of  the  curve  to  which  the  normals  belong. 

The  following  examples  will  further  illustrate  this  theory. 

EXAMPLES. 

(1 07. )  1 .  To  determine  the  curve  which  touches  an  infinite  series 
of  equal  circles,  whose  centres  are  all  situated  on  the  same  circum- 
ference. 

Let  the  equation  of  the  fixed  circle  be 

^2    ^    y>2   ^   ^'2^ 

then,  for  the  coordinates  of  the  centre  of  any  of  the  variable  circles, 
the  expressions  will  be 

x  and  \/r"^  —  x'^. 


148  THE  DIFFERENTIAL  CALCULUS. 

SO  that  the  general  representation  of  these  circles  will  be 


(x  _  x'f  +  (y—  Vr'-'  —  x'y—r"  ==  0  =  « (1), 

x'  being  considered  as  an  arbitrary  constant.  If,  however,  x'  be  con- 
sidered not  as  an  arbitrary  constant,  but  as  a  function  of  x  and  y, 

du 
fulfilling  the  condition  -y-,  =  0,  then,  by  the  preceding  theory,  (1) 
tix 

will  represent  the  curve  which  touches  all  the  circles  in  those  points 
where  each  is  intersected  by  its  consecutive  circle.  Hence,  differ- 
entiating (1)  with  respect  to  x,  we  have 

-^,  = —  {x —  x)-\-      — -  —  x=^ 

ax  ^  Vr  — X' 


...  _  X  Vr'^  —  x'^'+x'y  =  0 
'/x 

'''  ""'  ^  N/a^  +  / 

This,  then,  is  the  function  ofx,  y,  which,  substituted  for  x,  in  (1), 
gives  the  equation  of  the  locus  sought.  The  result  of  this  substitu- 
tion is 


x^  +  y^  —  2r'  Va;'  +  y"  +  r"  =  r", 
or,  extracting  the  root  of  each  side. 


>/a^  +  y^  =  r'  zL  r  .-.  x^  +  y^  —  {r  ±  ry, 

an  equation  representing  two  circles,  whose  radii  are  respectively 
r'  +  r  and  r  —  r.  Hence  the  series  of  circles  are  touched  and 
enveloped  by  two  circular  arcs,  having  these  radii,  and  the  same  centre 
as  the  fixed  circle. 

2.  Between  the  sides  of  a  given  angle  are  drawn  an  infinite 
number  of  straight  lines,  so  that  the  triangles  formed  may  all  have 
the  same  surface,  required  the  curve  to  which  every  one  of  these  lines 
is  a  tangent. 

Let  the  given  angle  be  6,  and,  taking  its  sides  for  axes,  we  have, 
for  the  equation  of  every  variable  line, 

T/  =  ax  +  /3  .  .  .   .   (1), 

and,  putting  successively  y  =  0  and  a?  =  0,  the  resulting  expressions 
for  X  and  y  denote  the  sides  of  the  variable  triangle,  including  the 


-4 1,  -t^— \— A-^W"  '■ 


THE    DIFFBRKNTIAL    CALGVLVS.  149 

given  emgle,  so  that  these  sides  are and  [3;  hence,  calling  the 

constant  surface  s,  we  have 

/S^    .  /S^sin.^ 

s  = sm.  e  .'.  a,  = ; 

2a  2s 

hence  the  equation  (1)  is  the  same  as 

y  =  —  "^-^^  x-\-^  .  .  .  .  (2), 

where  (3  is  considered  as  an  arbitrary  constant.     But  if  for  this  arbi- 
trary constant  we  substitute  the  function  of  a^,  arising  from  the  condition 

dy 
Tg-  =  0,  then  (2)  will  represent  the  locus  of  the  intersections  of 

each  variable  line,  with  its  consecutive  line,  which  locus  touches 
them  all.     Differentiating  them  with  regard  to  ^,  we  have 

{3  sin.  6       ,  ^  s 

— x+  1  =0.-.  ^= ^— , 

s  X  sm.  & 

this  substituted  in  (2)  gives^for  the  equation  of  the  sought  curve 

or  rather 

_         8 

hence  the  curve  is  an  hyperbola,  having  the  sides  of  the  given  angle 
for  asymptotes. 

3.  The  centres  of  an  infinite  number  of  equal  circles  are  all 
situated  on  the  same  straight  line  :  required  the  line  which  touches 
them  all  ? 

Ans.  They  are  touched  by  two  parallels  to  the  line  of 
centres. 

4.  From  every  point  in  a  parabola  lines  are  drawn,  making  the 
same  angle  with  the  diameter  that  the  diameter  makes  with  the  tan- 
gent :  required  the  hne  touching  them  all? 

Am.  They  are  touched  by  a  point,  viz.  the  focus,  in 
which  therefore  they  all  meet. 


150  THE  DIFFERENTIAL  CALCULUS. 


CEAFTER    IV. 

ON  THE  SINGULAR  POINTS  OF  CURVES,  AND  ON 
CURVILINEAR  ASYMPTOTES. 

jyitiltiple  Points. 

(108.)  If  several  branches  of  a  curve  meet  in  one  point,  whether 
by  intersecting  or  touching  each  other,  that  point  is  called  a  multiple 
point.  In  the  former  case  the  point  is  said  to  be  of  the  first  species, 
and  in  the  latter  of  the  second  species,  and  we  propose  here  to  in- 
quire how,  by  means  of  the  equation  of  any  curve,  these  points,  if  any, 
may  be  detected. 

JVfultiple  points  of  the  first  species.  When  the  curve  has  multiple 
points  of  the  first  species,  we  readily  arrive  at  the  means  of  determin- 
ing their  position  from  the  consideratioa  that  at  such  points  there  must 
be  as  many  rectilinear  tangents  as  there  are  touching  branches,  and, 

dii 
consequently,  as  many  values  for  ^,  the  tangent  of  the  inclination 

ax 

of  any  tangent  through  the  point  {x,  y)  to  the  axis  of  a- ;  so  that  the 

equation  of  the  curve  being  freed  from  radicals  and  put  under  the 

form 

F  {X,  y)  =  0, 

its  multiple  points  of  the  first  species  will  all  be  given  analytically 

by  the  equation 

, du   ,   du 0 

dx       dy       O' 

so  that  no  systems  of  values  for  x  and  y  can  belong  to  multiple  points 

of  the  fiirst  species,  but  such  as  satisfy  the  conditions 

dx  dy 

as  well  as  the  equation  of  the  curve.  Having,  therefore,  determined 
all  such  systems  of  values  by  solving  the  two  last  equations,  the  true 
values  of  p'  for  each  system  will  be  ascertamed  by  proceeding  as  in 
(41),  and  those  systems  only  will  belong  to  multiple  points  of  the 


THE  DIFFERENTIAL  CALCULUS.  151 

first  species  that  give  multiple  values  to  p'.     Let  us  apply  this  to  an 
example  or  two. 

EXAMPLES. 

(109.)     1.  To  determine  whether  the  curve  represented  by  the 
equation 

ay^  —  ar'y  —  bsP  =  0, 
has  any  intersecting  branches 

At  the  points  where  branches  intersect  we  must  have 
3^"  {y  +  b)  =  0,  'Saf  —  0^  =  0 
.'.  x  =  0,y  —  0 


X  =  y/  3ab%  y  =  —  6  ; 
this  second  system  of  coordinates  do  not  satisfy  the  proposed  equa- 
tion, and  therefore  do  not  mark  any  point  in  the  curve ;  the  first  sys- 
tem, which  is  admissible,  shows  that  if  there  exist  any  multiple  point 
it  must  be  at  the  origin.  Hence,  to  ascertain  the  true  value  of  p'  at 
this  point,  we  have,  by  differentiating  both  numerator  and  denomina- 
tor in  the  expression 

__    6x  (y  +  6)  +  Sx'p'    _  0 
"~  •-       6ayp'  —  3ar       -"  ~  0 

_  My  +  6)  +  12p'x  -\-  3x^p"   _       66  n=,y^ 

L       6ayp"  +  6ap'2  —  6x      ^      6a[ p'f"^^^      ^  a' 
therefore,  as  this  has  but  one  real  value,  the  curve  has  no  intersecting 
branches. 

2.  To  determine  whether  the  curve  represented  by  the  equation 
x'*  +  2a2fyf  —  ay^  =  0 
has  intersecting  branches 

^  =  4x{x^+ay)  =  0,~  =  a  {2x^-3^)  =  0. 


I 


152 


THE  DIFFERENTIAL  CALCULUS. 


There  is  but  one  system  of  values  that  can  satisfy  these  three  equa- 
tions, viz. 

so  that  if  there  are  intersecting  branches  they  must  intersect  at  the 
origin.  To  determine,  therefore,  whether  at  this  point  p'  has  multiple 
values  we  have 


[Pl 


4x  {jr  +  ay)  -,  _  0 

_    Gr"  +  2ay  +  2axp'    _  0 
'-       3ayp'  —  2ax     /  ~0 
-        4a  [p'] 


3a[p'Y  —  2a 
.:  3a  [p'Y  —  6a  [pq  "=  0 
...  [p']  =  0  or  [/]  =  ±  V  2; 
hence  three  branches  of  the  curve  intersect  at  the 
origin ;  the  tangent  to  one  of  them  at  that  point  is 
parallel  to  the  axis  of  x,  and  the  tangents  to  the 
other  two  are  symmetrically  situated  with  respect 
to  the  axis  of  ?/,  since  they  are  inclined  to  the  axis 
of  X,  at  angles  whose  tangents  are  -\-  V  2  and  —  y/  2. 

(110.)  Should  the  values  ofp'  corresponding  to  any  values  of  ar 
and  y,  which  satisfy  the  equation  of  the  curve,  be  all  imaginary,  we 
must  infer  that,  although  such  a  system  of  values  belong  to  a  point 
of  the  locus,  yet  that  point  must  be  detached  from  the  other  points  of 
the  locus,  for  since,  if  the  abscissa  of  this  point  be  increased  by  h, 
the  development  of  the  ordinate  will  agree  with  Taylor's  develop- 
ment, as  far,  at  least,  as  the  second  term  for  all  values  of  h,  between 
some  finite  value  and  0,  it  follows  that  all  the  corresponding  ordi- 
nates  between  these  limits  must  be  imaginary,  so  that  the  proposed 
point  is  isolated,  having  no  geometrical  connexion  with  the  curve, 
although  its  coordinates  satisfy  the  equation.  Such  a  point  is  called 
a  conjugate  point. 

(111.)  From  what  has  now  been  said,  it  appears  that,  by  having 
the  equation  of  a  plane  curve  given,  those  points  in  it  where  branches 
intersect,  as  also  those  which  are  entirely  detached  from  the  curve, 
although  belonging  to  its  equation,  may  always  be  determined  by  the 


THE    DIFFERENTIAL    CALCULUS.  153 

tipplication  of  the  differential  calculus,  and  independently  of  all  con- 
siderations about  the  failing  cases  of  Taylor's  theorem,  except,  in- 
deed, those  connected  with  the  theory  of  vanishing  fractions.  We 
shall  now  seek  the  analytical  indications  of 

JVLulti'ple  Points  of  the  Second  Species. 

(112.)  The  second  species  of  multiple  points,  or  those  where 
branches  of  the  curve  touch  each  other,  the  differential  calculus  does 
not  furnish  the  means  of  readily  determining  from  the  implicit  equa- 
tion of  the  curve.  We  know  that  at  such  a  point,  p'  cannot  admit  of 
different  values,  since  the  branches  have  one  common  tangent ;  and 
we  know,  moreover,  that  if  Taylor's  theorem  does  not  fail  at  that 
point,  we  shall,  by  successively  differentiating,  at  length  arrive  at  a 
coefficient  which,  being  put  under  the  form  ^,  the  different  values  will 
indicate  so  many  different  touching  branches  ;  for  if  no  coefficient  gave 
multiple  values  for  the  proposed  coordinates  x',  y,  then  the  ordinates 
corresponding  to  the  abscissas  between  the  limits  x  and  x  ±  h.,h 
being  of  some  finite  value,  would  each  have  but  one  value,  and,  there- 
fore, different  branches  could  not  proceed  from  the  point  {x\  if).  But 
we  have  no  means  of  ascertaining  «  priori  which  of  the  coefficients 
furnishes  the  multiple  value.  When,  however,  the  equation  of  the 
curve  is  explicit,  then  the  multiple  points  of  either  species  are  very 
easily  determined.     Thus,  if  the  equation  of  the  curve  be 


e^ 


y  ~  {x  —  ay  V  X  —  6  +  c, 
we  at  once  see  that  x  =  a  destroys  the  radical  in  y  and  p',  that  re-ap- 
pears in  p" ;  therefore,  at  the  point  corresponding  to  this 
abscissa,  these  will  be  but  one  tangent,  and  yet  two 
branches  of  the  curve  proceed  from  it  on  account  of  the 
double  value  ofp".  Hence  the  point  is  a  double  point  of  the  second  spe- 
cies, the  branches  have  contact  of  the  first  order,  and,  because  p'  =  0, 
the  common  tangent  is  parallel  to  the  axis  of  the  abscissas ;  if  the  radical 
had  been  of  the  third  degree,  the  point  corresponding  to  the  same  abscissa 
would  have  been  a  triple  point,  &c.  It  appears,  therefore,  that  when 
the  equation  of  the  curve  is  solved  for  y,  there  will  exist  a  multiple 
point,  if  in  the  expression  for  x  a  radical  is  multiplied  by  the  factor 
{x  —  a)"*.  If  ni  =  1,  the  branches  of  the  curve  intersect  at  the  point 
■whose  abscissa  is  x  =  a,  because  then  p'  at  that  point  takes  the  same 

20 


154  THE  DIFFERENTIAL  CALCULUS. 

values  as  the  radical,  but  if  m  >  1  then  the  branches  touch,  because 
then  the  radical  is  destroyed  in  p'  for  .t  =  a ;  in  both  cases  the  index 
of  the  radical  will  denote  the  number  of  branches  which  meet  ui  the 
point.  Such,  therefore,  are  the  geometrical  significations  of  the  cases 
discussed  in  (75)  and  (76). 

Ctisps,  or  Points  of  Regression. 

(113.)  A  cusp  or  point  of  regression  is  that  particular 

—  kind  of  double  point  of  the  second  species  in  which  the 
two  touching  branches  terminate,  and  through  which  they 

—  do  not  pass,  so  that  on  one  side  of  such  a  point,  viz.  on 
that  where  the  branches  lie,  the  ordinate  has  a  double 
value,  and  on  the  other  side  the  contiguous  ordinate  has 
an  imaginary  value. 

The  cusp  represented  in  the  first  figure,  where  the  branches  are 
one  on  each  side  of  the  common  tangent,  is  called  a  cusp  of  the  first 
kind,  and  that  in  the  second  figure,  where  the  branches  are  both  on 
one  side,  a  cusp  of  the  second  kind. 

(114.)  It  is  obvious  that  cusps  can  exist  only  at  those  points,  the 
particular  coordinates  of  which  cause  Taylor's  theorem  to  fail,  for  if 
Taylor's  theorem  did  not  fail  at  such  a  poini,  then  the  ordinates  in  the 
vicinity,  corresponding  both  to  x  +  h  and  to  a;  —  h,  would  be  both 
possible  or  impossible  at  the  same  time.  We  are  not,  however,  to 
infer  that  when  the  adjacent  ordinates  are  real  on  the  one  side  of  any 
point,  and  on  the  other  side  imaginary,  that  a  cusp  necessarily  exists 
at  that  point,  for  it  is  plain  that  the  same  analytical  indications  are 
furnished  by  the  point  which  limits  any  curve  in  the  direction  of  the 
axis  of  X,  or  at  which  the  tangent  is  perpendicular  to  that  axis,  as  in 
the  third  figure.  It  becomes  important,  therefore,  in  seeking  parti- 
cular points  of  curves  to  be  able  to  distinguish  the  point  which  limits 
the  curve  in  the  direction  of  the  axes  from  cusps. 

(115.)  Now  at  the  limits,  the  tangents  to  the  curve  are  parallel  to 

the  axes,  the  limits  are  therefore  determined  by  the  equations  -^  =  ao 

ax 

and  ^  —  ^1  ^^^  *h®y  fulfil,  moreover,  the  following  additional  con- 


THE  DIFFERENTIAL  CALCULUS.  155 

ditions,  viz.  1°,  the  ordinate  or  abscissa,  whichever  it 


^—^ 


may  be,  that  is  parallel  to  the  tangent,  immediately  be- 
yond the  limit,  must  be  imaginary ;  but  if  it  be  ascertained 
that  this  is  not  the  case,  the  point  is  not  a  limit  but  a  cusp 
of  the  first  kind,  posited  as  in  the  annexed  figures,  or 
else  a  point  of  inflexion ;  the  latter  when  the  contiguous 
ordinates  are  the  one  greater  and  the  other  less  than  that  at  the     | 
point.     2°,  Besides  the  first  condition  there  must  exist  also    i^//' 
this,  viz.  that  immediately  ivithin  the  limit  the  double  ordinate     ' 
or  abscissa,  whichever  may  be  parallel  to  the  tangent,  must 
have  one  of  its  values  greater  and  the  other  less  than  at  the 
point,  but  if  both  are  greater  or  both  less  the  point  is  not  a  limit 
but  a  cusp  of  the  second  kind,  posited  as  in  the  annexed  figures. 
Hence,  when  the  branches  forming  the  cusp  touch  the  abscissa  or  the 
ordinate  of  the  point,  they  may  be  discovered  by  seeking  among  the 

values  which  satisfy  the  equations  -j^  =  0  and  -^^  =00  ,  those  which 

(XJ[/  (XX 

do  not  fulfil  both  the  foregoing  conditions.     Let  us  illustrate  this  by 
examples. 

EXAMPLES. 

(116.)     1.  To  determine  whether  the  curve  whose  equation  is 

{y  —  bf={x-aY 

has  a  cusp  at  the  point  where  the  tangent  is  parallel  to  the  axis  of?/. 
By  differentiating 

dy  _  2         X  —  a 

'dx~2,'      {y  —b  y 
this  becomes  infinite  for  y  =  b,  therefore  the  point  to  be  examined  is 
(a,  6).  In  order  to  this,  substitute  a  ±:  h  for  x,  in  the  proposed  equa- 
tion, and  we  have,  for  the  contiguous  ordinates, 

y  =  b±  h^, 
which  is  not  imaginary  either  for  +  ^  or  —  A ;  the  point 
(a,  b)  is  therefore  a  cusp  of  the  first  kind,  and  posited  as 
in  the  figure,  since  the  contiguous  values  of  y  are  both 
greater  than  b. 

2.  To  determine  whether  the  curve  whose  equation  is 


156  THE    DIPFERENTIAL    CALCULtlS. 

tj  —  a  =  {x  —  b)^-\-{x—  b)* 
has  a  cusp  at  the  point  where  the  tangent  is  parallel  to  the  axis  of  if. 

Here  the  coefficient  -~  becomes  infinite  for  x  =  b,  therefore  the 
ax 

point  to  be  examined  is  (6,  a).     Substituting  6  +  ft  for  x,  we  have 

y  =  a  +  h^  -\-  h*. 
For  negative  values  of  h  this  is  imaginary,  therefore  the  curve  lies 
entirely  to  the  right  of  the  ordinate  y  =  a,  so  that  the  condition  1° 
pertaining  to  a  Hmit  is  fulfilled.  To  the  right  of  this  or- 
dinate the  two  values  of  y,  corresponding  to  a  value  of  ft 
ever  so  small,  are  both  greater  than  y  =  a,  so  that  the 
condition  2°  is  not  fulfilled,  the  point  (6,  a)  is  therefore 


a  cusp  of  the  second  kind,  and  posited  as  in  the  cut. 

3.  To  determine  the  point  of  the  curve  whose  equation  is 

{y  —  a-xy  =  {x  —  by, 

at  which  the  tangent  is  parallel  to  the  axis  of  y. 

The  differential  coefficient  becomes  infinite  for  x  =  b,  therefore 
the  point  to  be  examined  is  (6,  a  +  6).     Substituting  b  -\-  h  for  a:, 

2/  =  (a  +  6)  +  ft*  +  ft, 

^       negative  values  of  ft  render  this  imaginary,  therefore  the 
*\^       condition  1°  is  .fulfilled;  positive  values  give  two  values 

3, 

for  y,  and  as  ft  may  be  taken  so  small  that  ft*  may  exceed 

3 

ft,  and  since,  moreover,  the  two  values  of  ft*  are  the  one  positive  and 
the  other  negative,  it  follows  that  the  real  ordinate  contiguous  to  the 
point  has  one  value  greater,  and  the  other  less,  than  that  at  the  point 
of  contact ;  hence  the  condition  2°  is  also  fulfilled,  and  thus  the  point 
marks  the  limit  of  the  curve,  which,  therefore,  lies  to  the  right  of  the 
ordinate,  through  x  =  b. 

(117.)  Having  thus  seen  how  to  determine  those  cusps  where  the 
branches  touch  an  ordinate  or  abscissa,  we  shall  now  seek  how  to 
discover  those  at  which  the  tangent  is  oblique  to  the  axes.  The  true 
development  of  the  ordinate  contiguous  to  such  a  cusp  must  be  of  the 
form 

y'^'^h^  Aft"+Bft^  +  &c. 
ax 


THE  DIFFERENTIAL  CALCULUS. 


157 


and  the  corresponding  ordinate  of  the  tangent  will  be 

hence,  subtracting  this  from  the  former,  we  have 

A  =  A/i"  +  /3/i^  +  &c. 

(118.)  Now  in  order  that  the  pouxt  (x',  y')  may  be  a  cusp,  this  dif- 
ference for  a  small  value  of  h  must  have  two  values,  and  to  be  a  cusp 
of  the  first  kind  these  two  values  must  obviously  have  opposite  signs  ; 

but  since  h  may  be  so  small  that  A^    may  exceed  the  sum  of  all  the 

following  terms,  h  must  have  two  opposite  values ;  hence,  a  must 
be  a  fraction  with  an  even  denominator,  and,  conversely,  if  a  be  a 
fraction  with  an  even  denominator,  the  point  {x',  y')  will  be  a  cusp  of 

the  first  kuid.     Hence,  at  such  a  point,  -7^  is  either  0  or  oo  :  0  if 

/3  >2,  and  00  if^  <  2. 

(119.)  In  order  that  the  cusp  may  be  of  the  second  kind,  both 

values  of  A  must  have  the  same  sign  there,  for  A  cannot  admit  of 
opposite  values  of  the  same  value  of  h,  consequently  a  must  in  this 
case  be  either  a  whole  number,  or  else  a  fraction  with  an  odd  denomi- 
nator ;  and  conversely,  if  a  be  either  a  whole  number,  or  a  fraction 
with  an  odd  denominator,  the  point  (x,  y')  will  be  a  cusp  of  the  second 
kind,  provided,  of  course,  that  A  has  two  values.  The  position  of  the 
branches  will  depend  on  the  sign  of  A. 

We  shall  now  give  an  example  or  two. 

(120.)     4.  To  determine  whether  the  curve  whose  equation  is 

y  =^  X  ±  x^ 
has  a  cusp. 

Here  y  is  possible  for  positive  values  of  x,  and  imaginary  for  all 
negative  values  ;  hence  there  may  he  a  cusp  at  the  origin.  To  as- 
certain this,  put  h  for  x,  in  the  equation,  and  we  have,  for  the  con- 
tiguous ordmate,  the  value 

y  =  h  ±  h^. 


I 


158  THE    DIFFERENTIAL    CALCULUS. 

/^X  The  coefficient  of /i  being  1  =  -j-,  we  see  that  the  tan- 

gent to  the  curve  at  the  origin  is  inclined  at  45°  to  the 
axes,  and,  since  \  has  an  even  denominator  the  origin  is  a  cusp  of  the 
first  kind. 

6.  To  determine  whether  the  curve  whose  equation  is 

5 

y  —  a  =  a?  +  hi?  +  cx^ 
has  a  cusp. 

Here  ?/  is  imaginary  for  all  negative  values  of  .r,  therefore  the  point 
(0,  a)  nmy  be  a  cusp.     Substituting  h  for  x,  we  have 

t/  =  o  +  /i  +  i/t'  +  cli^. 
1^/  As  before,  the  tangent  is  incUned  at  45°  to  the  axes, 
and,  since  the  exponent  of  the  third  term  is  a  whole  num- 
ber,  and  the  whole  expression  admits  of  two  values,  in 


consequence  of  the  even  root  \i^ ,  it  follows  that  the  proposed  point 
is  a  cusp  of  the  second  kind.  The  branches  are  situated  to  the  right 
of  the  axis  of  ?/,  because  h  must  be  positive,  and  they  are  above  the 
tangent  because  h]^  is  positive. 

6.  To  determine  whether  the  curve  whose  equation  is 
(2y  +  .T  +  \f=  2(1— x)** 
has  a  cusp. 

Here  values  of  x  greater  than  1  are  obviously  inadmissible,  and 
to  this  value  of  x  corresponds  ?/  =  —  1 ;  hence  the  point  having  these 
coordinates  may  be  a  cusp.     Substituting  1  +  fe  for  x,  we  have 

therefore  the  tangent  to  the  curve  at  the  proposed  point  has  the  tri- 
gonometrical tangent  of  its  inclination  to  the  axis  of  x  equal  to  \ ,  and 
since  the  fraction  f  has  an  even  denominator,  the  point 
is  a  cusp  of  the  first  kind.  Because  /i  is  negative,  the 
branches  are  to  the  left  of  the  ordinate  to  the  point  which 
\\       is  below  the  axis  of  x,  because  this  ordinate  is  negative. 

Points  of  Inflexion. 
(121.)  Points  of  inflexion  have  been  defined  at  (92),  and  we  have 


THE  DIFFERENTIAL  CALCULUS.  159 

there  shown  that  a  point  of  this  kind  always  exists  when  its  abscissa 
causes  all  the  differential  coefficients  to  vanish  between  the  first  and 
the  nih,  provided  the  nth  be  odd  and  becomes  neither  0  nor  go  .   The 

simplest  indication  therefore  of  a  point  of  inflexion  is  [-rj]  —  0,  and 

[  -fj^]  neither  0  nor  co  ;  such  indications,  however,  cannot  be  fur- 
nished by  any  point  at  which  the  tangent  is  parallel  to  the  axis  oft/, 
since  in  this  case  [y^]  and  all  the  following  coefficients  become  infi- 
nite. Neither  can  these  indications  take  place  at  any  point,  for  which 
Taylor's  theorem  fails  after  the  third  term.  It  becomes,  therefore, 
of  consequence,  in  examining  particular  points  of  a  curve,  to  be  able 
to  detect  the  existence  of  points  of  inflexion  by  some  general  method, 
independently  of  the  diftcrential  coefficients  beyond  the  first.  The 
only  general  method  of  doing  this  is  that  which  we  have  already  em- 
ployed for  the  discovery  of  cusps,  and  which  consists  simply  in  ex- 
amining the  course  of  the  curve  in  the  immediate  vicinity  and  on  each 
side  the  point  in  question.  Points  of  inflexion  are  somewhat  similar 
to  cusps,  each  having  some  of  the  analytical  characteristics  common 
to  both,  and  to  the  limiting  points  of  curves  as  already  hinted  at  in 
(114).  But  the  characteristic  property  of  a  point  of  inflection  is,  that 
the  adjacent  ordinates  on  each  side  are  the  one  greater  and  the  other 
less  than  the  ordinate  at  the  point.  This  pecuharity  distinguishes  a 
point  of  inflexion  from  a  limit,  inasmuch  as  at  a  limit 
the  ordinate  immediately  beyond  is  imaginary ;  and 
it  distinguishes  it  from  a  cusp  of  the  first  kind,  in- 
asmuch as  at  such  a  cusp  the  adjacent  ordinates 
are  either  both  greater  or  both  less  than  at  the  point, 
or  else,  as  is  the  case  when  the  tangent  at  the  point  is  oblique  to  the 
axes,  one  of  these  ordinates  is  imaginary,  the  other  double.  We  have 
then  first  to  ascertain  at  what  points  of  the  curve  inflexions  may  ex- 
ist, or  to  find  what  points  are  given  by  the  conditions 

^  =  Q   =Oorcc, 

or,  which  is  the  same  thing,  what  points  are  given  by  the  separate 
conditions. 


160  THE  DIFFERENTIAL  CALCULUS. 

P  =  0,  Q  =  0, 
we  are  then,  by  examining  the  course  of  the  curve  in  the  vicinity  of 
each  point,  to  determine  to  which  of  them  really  belongs  the  charac- 
teristic of  an  inflexion. 

Thus  the  means  of  distinguishing  points  of  inflexion  being  sufli- 
ciently  clear,  we  shall  proceed  to  a  few  examples. 

EXAMPLES. 

1.  To  determine  whether  the  curve  whose  equatio  n 

y  =  b  +  {x  —  ay 
has  a  point  of  inflexion  where  the  tangent  is  parallel  to  the  axis  of  a:. 
Here 

p'  =zB(^x  —  ay, 
and  when  the  tangent  is  parallel  to  the  axis  oi"  x,  p'  =  0, .:  x  =  a 
and  2/  =  6,  at  the  proposed  point.     In  the  vicinity  a;  =  a  +  A, 

...  ij  =  b  +  /i^ 
which  is  greater  than  b,  the  ordinate  of  the  point  when  h  is  positive, 
and  less  when  h  is  negative  ;  the  point  (a,  b)  is  therefore  a  point  of 
inflexion. 

2.  To  determine  whether  the  curve  whose  equation  is  y'^  =  x^  or 

5. 

y  =  x^  has  an  inflexion  at  any  point. 

2  1. 

3  *    »   i^  3*3* 

this  becomes  co  for  x  =  0,  therefore  a  point  of  inflexion  may  exist 
at  the  origin.     Putting  h  for  x  we  have 

y  =  h^, 

which  is  greater  than  0,  the  ordinate  of  the  point,  when  h  is  positive, 
and  less  when  h  is  negative ;  hence  there  is  an  inflexion  at  the  ori- 
gin.     Also  the    equation  of   the  tangent  being 

?/  =  -I  x^,  the  ordinates  corresponding  to  x  =  ±  h 
are  both  less  than  those  given  by  the  above  equa- 
tion ;  hence  the  curve  lies  above  the  tangent  to  the 
right  of  the  origin,  and  below  it  to  the  left,  as  in  the 
figure. 

3.  To  determine  whether  the  curve  whose  equation  is 


J 


THE  DIFFERENTIAL  CALCULUS.  161 

y  —  X  =  (x  —  a)^ 
has  a  point  of  inflexion 

p'  =  i  +  f  (a:  — «)^p"=l•f  (^  — «)"' 
this  becomes  infinite  for  x  =  a,  therefore  a  point  of  inflexion  nwy 
exist  at  the  point  (a,  a).     In  the  vicinity  of  this  point  x  =  a  +  ^i 

.5. 

.♦.  1/  =  «  +  /i  +  h^t 
which  is  greater  than  a  when  h  is  positive,  and  less 
when  h  is  negative ;  hence  (a,  a,)  is  a  point  of  in- 
flexion. As  the  corresponding  ordinates  of  the 
tangent  y  =  a  ±  h,  one,  viz.  y  =  a  -^  h,  is  less 
than  that  of  the  curve,  and  the  other  greater ;  hence 
the  curve  bends,  as  in  the  figure. 

On  Curvilinear  Asymptotes. 

(122.)  Two  plane  curves,  having  infinite  branches,  are  said  to  be 
asymptotes  to  each  other,  when  they  approach  the  closer  to  each 
other  as  the  branches  are  prolonged,  but  meet  only  at  an  infinite  dis- 
tance.* 

Hence,  since  the  expression  for  the  difference  of  the  ordinates  cor- 
responding to  the  same  abscissa  in  two  such  curves  becomes  less 
as  the  abscissa  becomes  greater,  and  finally  becomes  0,  when  the 
abscissa  becomes  co  ,  it  follows  that  that  expression  can  contain 
none  but  negative  powers  of  x,  without  the  addition  of  any  con- 
stant quantity.  For,  if  a  positive  power  of  x  entered  the  expres- 
sion for  the  difference,  that  expression  would  become  not  0  but  oo  , 
when  X  =  CO  ;  and,  if  there  were  a  quantity  independent  of  x,  the  dif- 
ference would  be  reduced  to  this  quantity,  and  not  to  0,  for  x  =  0, 
Hence  two  curves  are  asymptotes  to  each  other,  when  the  general 
expression  for  the  difference  of  the  ordinates  corresponding  to  the 
same  abscissa  is 

A  =  A'x~"  +  B'x"^  +  C'x"y  +  &c (1), 

or  when  the  general  expression  for  the  difference  of  the  abscissas  cor- 

*  Spirals  meet  their  asymptotic  circles  only  after  an  infinite  number  of  revolu- 
tions ;  these  we  do  not  consider  here,   having  examined  them  at  (85). 

21 


162  THE  DIFFERENTIAL  CALCULUS. 

responding  to  the  same  ordinate  is 

A  =  A'y~"'  -\-  B'y"^  +  C'y~^  +  &c (2), 

and  conversely,  when  the  curves  are  asymptotes  to  each  other  ;  one 
or  both  these  forms  must  have  place. 

If  for  one  of  the  curves  whose  corresponding  ordinates  are  sup- 
posed to  give  the  difference  (1)  there  be  substituted  another,  which 
would  reduce  that  difference  to 

B'x~l^  +  C'x~'y  +  &c. 

this  new  curve  would  be  an  asymptote  to  both,  and  would  obviously, 
throughout  its  course,  continually  approach  nearer  to  that  which  it 
has  been  compared  to,  than  the  one  for  which  we  have  substituted  it 
does.  In  like  manner,  if  a  third  curve  would  further  reduce  the  dif- 
ference (1)  to 

c'x~y  +  &c. 

this  third  curve  would  approach  the  first  still  nearer,  and  all  the  four 
would  be  asymptotes  to  each  other.  It  appears,  therefore,  that  every 
curve  of  which  the  ordinate  may  be  expanded  into  em  expression  of 
the  form 

y  =  Aa^  +  Bx*  +  .  .  .  .  A'x~"-  +  B'x~^  +  &c (3). 

admits  of  an  infinite  number  of  asymptotes. 

Since  the  general  expression  for  the  ordinate  of  a  straight  line  is 
y  =  Ax  +  B,  for  the  difference  between  this  ordinate  and  that  of  a 
curve  at  the  point  whose  abscissa  is  x,  to  have  the  form  (1),  the  equa- 
tion of  the  curve  must  be 

y  =  Ax  +  B+  A'x'"-  +  B'x~^  +  &c (4), 

this  equation,  therefore,  comprehends  all  the  curves  that  have  a  rec- 
tilinear asymptote,  and  among  them  the  common  hyperbola,  whose 
equation  is 

y=±^{jp  —  Ay  =   zp  -?  a:  q=  i  ABx"'  +  &c. 

The  curves  included  in  the  equation  (4)  are  therefore  called  hyper- 
bolic curves. 

The  other  curves  comprised  in  the  more  general  equation  (3),  not 
admitting  of  a  rectilinear  asymptote,  are  c^}ied  parabolic  curves. 


THE  DIFFERENTIAL  CALCULUS. 


i6d 


The  common  hyperbola  we  see  by  the  above  equation  admits  of 

TJ 

the  two  rectilinear  asymptotes  y  =  ±  j-  x,  and  of  an  infinite  num- 
ber of  hyperbolic  asymptotes. 

As  an  example  of  this  method  of  discovering  rectilinear  and  cur- 
viUnear  asymptotes,  let  the  equation 

my'^  —  xy^  =  mx^ 
be  proposed.     The  development  of  y  in  a  series  of  descending  powers 
ofx  is  (Ex.  9,  p.  62,) 

y  =  —  m r  —  &c. 

•^  x^ 

therefore  the  curve  has  one  rectilinear  asymptote,  parallel  to  the  axis 

of  X,  its  equation  being  y  =  —  wi ;  the  hyperbolic  asymptote  next  to 

this,  and  which  lies  closer  to  the  curve,  is  of  the  fourth  order,  its 

equation  being 

yx"^  +  mx"^  +  m''  =  0. 

Again,  let  the  equation  of  the  proposed  curve  be 

b 

y  ~ 

=  bx-'  +  &c (1), 

also,  since 

b^  b^ 

x^  ->  a^  =  —  .-.  .r  =  a  +  J- .  —  «-^  +  &c (2). 

if  ^     a  ^ 

From  (I)  it  appears  that  the  curve  has  a  rectilinear  asymptote,  coin- 
cident with  the  axis  of  x,  its  equation  being  ?/  =  0  ;  the  hyperbola 
whose  asymptotes  coincide  with  the  axes  is  also  an  asymptote,  its 
equation  being  xy  =  6.  From  (2)  it  appears  that  the  curve  has 
another  rectilinear  asymptote,  parallel  to  the  axis  ofy,  its  equation 
being  x  =  a  ;  the  hyperbola  next  to  this  is  of  the  third  order.  If  we 
consider  the  radical,  in  the  proposed  equation,  to  admit  of  either  a 
positive  or  a  negative  value,  then  there  will  be  two  rectilinear  asymp- 
totes, parallel  to  the  axis  of  y  and  equidistant  from  it,  as  also  two  hy- 
perbolic asymptotes,  symmetrically  situated  between  the  axes. 


164  THE  DIFFERENTIAL  CALCULUS, 


SECTION    III. 

ON  THE  GENERAL  THEORY  OF  CURVE  SURFACES 
AND  OF  CURVES  OF  DOUBLE  CURVATURE. 


CnilFTEH   I. 
ON  TANGENT  AND  NORMAL  PLANES. 

PROBLEM    I. 

(123.)  To  determine  the  equation  of  the  tangent  plane  at  any  point 
on  a  curve  surface. 

Let  (x'j  y\  z',)  represent  any  point  on  a  curve  surface  of  which  the 
equation  is 

z  =  F  {x,  y), 
then  the  tangent  plane  will  obviously  be  determined,  when  two  linear 
tangents  through  this  point  are  determined.     Let  us  then  consider, 
for  greater  simplicity,  the  two  linear  tangents  respectively  parallel  to 
the  planes  ofxz,  zy ;  their  equations  are 

z  —  z'  =  a{x  —  x')  \  ^ij^ 


} (2), 


y  =  y 

and 

z  —  z'  =  b{y  —  y') 
X  =  x' 

But  since  these  are  tangents  to  the  plane  curves,  which  are  the  sec- 
tions through  {x',  y',  z\)  parallel  to  the  planes  ofxz,zy,  therefore  (77) 
dz  ,   ,        dz' 

Moreover  the  traces  of  the  plane  through  the  lines  (1),  (2),  upon  the 
planes  of  xz,  zy,  being  parallel  to  the  lines  themselves,  a  and  b  must 
be  the  same  in  the  traces  as  in  these  lines,  and  since  they  are  the 


THE  DIFFERENTIAL  CALCULUS.  165 

same  in  the  plane  as  in  its  traces,  it  follows  that  the  equation  of  this 
plane  must  be 

z-2'=p'(x—x')  +  q'{y-y')  ....  (3), 
in  which  the  partial  differential  coefficients  p',  q',  express  the  trigono- 
metrical tangents  of  the  inclinations  of  the  vertical  traces  to  the  axes 
of  X  and  y  respectively. 

For  the  angle  which  the  horizontal  trace  makes  with  the  axis  of  a? 
we  have,  by  putting  s  =  0,  in  (3), 

P' 
tan.  mc.  ^. 

(124.)  If  the  equation  of  the  surface  is  given  under  the  form 
u  =  F{x,y,z,)=0  .  .  .  .  (4), 
then  the  expressions  for  the  total  differential  coefficients  derived  from 
tt,  considered  as  a  function,  first  of  the  single  variable  x,  and  then  of 
the  single  variable  y,  are  (57) 

.du^  du    .du    ,  _  f. 

dx         dx       dz 
,du    _  du   .du    ,  _  - 
^dy^       dy       dz^ 
from  which  we  get  the  values 

du  du 

, dx     ,  _       dy 

du  du 

dz  dz 

hence,  by  substituting  these  expressions  in  (3),  the  equation  for  the 
tangent  plane  becomes 

/v  <^M   ,    ,  ,.  du   ,    ,  ,.   du       ^  ,^. 

(.-.') 5^+  (.-.■)  3J+  fa-*')  ^  =  0  .  .  .  .  (5). 

PROBLEM    II. 

(125.)  To  determine  the  equation  of  the  normal  line  at  any  point 
of  a  curve  surface. 

We  have  here  merely  to  express  the  equation  of  a  straight  line, 
perpendicular  to  the  plane  (3),  and  passing  through  the  point  of  con- 
tact (ar',  y\  z'.) 


^f'' 

+  q" 
-q' 

+  1 

Vf' 

+  9" 

1 

+  1 

16B  THE  DIFFERENTIAL  CALCULUS. 

New  the  projections  of  this  line  must  be  perpendicular  to  the  traces 
of  the  tangent  plane,  or  to  the  lines  (1),  (2,)  hence  the  equations  of 
these  projections  must  be 

x  —  x'+p'{z  —  z')=0\ 

y  —  y'  +  q'{^  —  ^')  =  o] 

which  together,  therefore,  represent  the  normal. 

(126.)  If  we  represent  by  a, /3,  y,  the  inclinations  of  this  line  to 
the  axes  of  x,  y,  s,  respectively,  then  {Anal.  Geom.) 

cos.  a 


COS.  ^  = 


COS.  y       ^^,2  +  g'2  4-  1 

(127.)  If  the  equation  of  the  surface  be  given  under  the  form  (4), 
last  problem,  then,  in  these  expressions  for  the  inclinations,  we  must, 
instead  of  p'  and  q',  write  their  values  as  before  determined  from  that 
equation.     If,  for  brevity,  we  put 

_  1 

the  expressions  for  the  cosines  will  then  be 

du  -         du  du 

cos.  a  =  V  -r-,  COS.  p  =  1)  -r-,  COS.  y  =  t)  -j-. 

dx  dy  dz 

As  every  plane  which  contains  the  normal  Une  must  be  perpendicular 
to  the  tangent  plane,  it  is  obvious  that  there  exists  an  infinite  number 
of  normal  planes  to  any  point  of  a  surface. 

PKOBLEM    III. 

(128.)  To  determine  the  equation  of  the  tangent  line  to  any  point 
of  a  curve  of  double  curvature. 

We  have  already  indicated  {Anal.  Geom.)  how  this  equation  is  to 
be  determined : 

Let 


THE  DIFFERENTIAL  CALCULUS.  167 

y=fx,Z  =  Fx   ....    (1) 

be  the  equations  of  the  projections  of  the  proposed  curve,  on  the 
planes  o£xy,  xz,  and  let  (»',  y',  z',)  be  the  point  to  which  the  linear 
tangent  is  to  be  drawn,  which  point  will  be  projected  into  {x,  y')  emd 
{x',  z',)  on  the  plane  curves  (1),  therefore  tangents  through  them  to 
these  plane  curves  will  be  represented  by  the  two  equations 

these,  therefore,  together  represent  the  required  tangent  in  space. 

PROBLEM    IV. 

(129.)  To  determine  the  equation  of  the  normal  plane  at  any  point 
in  a  curve  of  double  curvature. 

The  equation  of  any  plane  passing  through  a  proposed  point  is 
{Anal.  Geom.) 

A{x  —  x')-\-B{y-y')  +  C{z  —  z')=0  ....  (1), 
and  for  the  traces  of  this  plane  on  the  planes  of  xy,  xz,  we  have,  by 
putting  in  succession  z  =  0,  y  =  0,  the  equations 

A  C 

z  —  z'  =  —-^{x  —  x)-)r-^y', 

but  since  these  two  traces  are  respectively  perpendicular  to  those 
marked  (2),  last  problem, 

B         ,  C  _    , 

hence  the  equation  (1)  becomes 

x  —  x'+p'  {y  _  7/')  +  5'  (5  —  2')  =  0  .  .  .   .  (2), 
which  represents  the  normal  plane  sought. 


168  THE  DIFFERENTIAL  CALCULUS^ 


GBAPTER  ZX. 

ON  CYLINDRICAL  SURFACES,  CONICAL  SURFACES, 
AND  SURFACES  OF  REVOLUTION. 

(130.)  These  surfaces  have  been  considered  in  the  Analytical 
Geometry,  and  the  general  equations  of  the  two  first  classes  have 
been  deduced,  on  the  hypothesis  that  the  directrix  is  always  a  plane 
curve.  We  shall  now  suppose  the  directrix  to  be  any  curve  situated 
in  space,  and  investigate  the  differential  equations  of  these  surfaces, 
as  also  ©f  surfaces  of  revolution  in  general. 

Conical  and  Cylindrical  Surfaces. 

PROBLEM    I. 

To  determine  the  equation  of  cylindrical  surfaces  in  general. 
Let  the  equations  of  the  generating  straight  line  be 

"~  X  =  az  -]-  a  \  (  a  =  X  —  az  ,-v 

y  =  bz+  ^]    •*•     \^=y  —  bz''''  ^*^' 

and  the  equations  of  any  curve  in  space  considered  as  the  directrix, 
F{x,y,z)=0,f{x,y,z)=0  ....  (2). 
Now  for  every  point  in  this  directrix,  all  these  equations  exist 
simultaneously ;  moreover,  the  constants  a,  6,  are  fixed,  since  the 
inclination  of  the  generating  line  does  not  vary,  but  the  constants 
a,  ^,  are  not  fixed,  since  the  position  of  the  generating  lines  does 
vary.  If,  then,  we  eUminate  x,  y,  z,  from  the  above  equations,  there 
will  enter,  in  the  resulting  equation,  only  the  constants  a,  6,  and  the 
indeterminates  a,  /3,  hence,  solving  this  equation,  for  /3  we  shall  get 
a  result  of  the  form  (3  =  (pa;  consequently,  if  we  now  substitute  in 
this  the  values  of  a  and  /3  given  above,  in  terms  of  x,  y,  z,  we  shall 
have  this  general  relation  among  these  variables,  viz. 

y  —  bz  =  cp  (x  —  az)  =  0  .  .  .  .  (3), 
which  is  the  equation  of  cylindrical  surfaces  in  general,  the  function 
<p  depending  entirely  on  the  nature  of  the  directrix. 


THE  DIFFERENTIAL  CALCULUS.  169 

(131.)  Now,  by  differentiation,  this  function  may  be  eliminated 
(68),  hence, 

—  ^P'   _  ^  —  "P' 
1  —  bq'        — aq'^ 

.♦.  op'  +  bq'  =  I  or  a -^  +  b  —-  =  I  ■■•  '  (4), 
^  ^  dx  ay 

which  is  the  general  differential  equation  of  cylindrical  surfaces. 

(132.)  The  same  equation  may  be  immediately  deduced  from  the 

general  equation  of  a  tangent  plane,  to  the  cylindrical  surface.  Thus, 

the  equation  of  any  tangent  plane,  through  a  point  {x',  y',  z)  being 

z  —  z'  =  p'{x  —  x')  +  9'  (?/  —  y'), 

the  condition  necessary  for  it  to  be  always  tangent  to  the  cylinder  on 
which  this  point  is  situated,  is  merely  that  it  may  be  always  parallel 
to  its  generatrix  (1),  and  this  condition,  expressed  analytically,  is 
{Anal.  Geom.) 

ap'  +  bq'  —I  =  0 (4), 

this  is,  therefore,  the  relation  which  must  have  place  between  the  par- 
tial differential  coefficients  derived  from  the  equation  of  the  surface, 
in  order  that  that  surface  may  be  cylindrical,  and  it  agrees  with  the 
relation  before  estabhshed. 

If,  in  this  equation,  we  write  for  p',  q',  their  values  deduced  from 
the  equation  m  =  0  of  any  cyhndrical  surface,  as  exhibited  in  (124), 
it  becomes 

f?M    ,    .  du       du 

dx  dy       dz 

PROBLEM  II. 

(133.)  Given  the  equation  of  the  generatrix,  to  determine  the  cy- 
lindrical surface  which  envelopes  a  given  curve  surface. 

Since  the  cylinder  envelopes  the  given  surface,  the  curve  of  con- 
tact is  common  to  both,  therefore  every  tangent  plane  to  the  cylinder 
touches  the  enveloped  surface  in  that  curve.  The  equation  of  any  of 
these  tangent  planes  is 

z  —  z'=p'(x  —  x')  +  q'  {y  —  y'), 
whether  p'  and  q'  be  derived  from  the  equation  of  the  surface,  and 
take  those  particular  values  which  restrict  them  to  the  curve  of  con- 

22 


1'70  THE  DIFFEKENTIAL  CALCULUS. 

tact,  or  whether  p'  and  q'  be  derived  from  the  equation  of  the  cylin- 
der, and  preserve  their  general  values,  because  in  the  one  case  the 
contact  of  each  tangent  is  confined  to  a  point  in  the  curve  of  contact, 
and  in  the  other  case  the  contact  extends  along  the  whole  length  of 
the  cylinder.  Hence,  for  the  curve  of  contact,  the  condition  (6)  must 
have  place,  as  well  as  for  the  entire  surface  of  the  cylinder.  The 
mode  of  solution  is,  therefore,  obvious  ;  we  must  deduce  p'  and  q 
from  the  equation  of  the  given  surface,  and  substitute  them  in  (6), 
the  result  combined  with  the  equation  of  the  given  surface,  will  ob- 
viously represent  the  curve  for  which  p'  and  q'  are  common  to  both 
surfaces ;  that  is  to  say,  we  shall  thus  have  the  equations  of  the  di- 
rectrix, and  that  of  the  generatrix  being  also  given,  the  particular  cy- 
lindrical surface  becomes  determined. 

(134.)  If  the  proposed  curve  surface  be  of  the  second  order,  then 
the  equation  (6)  will  necessarily  be  of  the  first  degree  in  x,  y,  2,  and 
will,  therefore,  represent  a  plane ;  so  that  the  combination  of  this, 
with  the  equation  of  any  surface,  must  necessarily  represent  a  plane 
section  of  that  surface ;  we  infer,  therefore,  that  if  any  cylindrical 
surface  circumscribe  a  surface  of  the  second  order,  the  curve  of  con- 
tact will  always  be  a  plane  curve,  and  consequently  of  the  second 
order,  and  therefore  the  cylinder  itself  must  be  of  the  second  order. 

PROBLEM  III. 

(135.)  To  determine  the  general  equation  of  conical  surfaces. 

Let  («', »/',  s',)  be  the  vertex  of  the  conical  siyrface,  then  since  the 
generatrix  always  passes  through  this  point,  its  equations,  in  any  po- 
sition, will  be 

x  —  a;'  =  a{z  —  z')\  ,,. 

0  J  —  ^^^ 


y  —  y'  =  b{z  —  z' 


Z 2  2  2 

Also  let  the  equations  of  the  directrix  be 

F{x,y,z)  =  Oj{x,y,z)  =0  .   .  .  .  (2), 
then,  since  for  every  point  in  this  line,  the  equations  (1)  and  (2)  ex- 
ist together,  we  may  eliminate  the  variables  x,y,  z;  the  result  will  be 
an  equation,  containing  the  fixed  constants  x',  y',  z'  and  the  indetermi- 
nates  a,  6 ;  therefore,  solving  this  equation  for  b,  we  shall  have  6  = 


THE    DIFFERENTIAL    CALCULUS.  171 

(pa.     Hence,  substituting  for  a,  b,  their  values  in  terms  of  x,  y,  we 
have 

y  —  y  x  —  x' 

7,  -  <P  ^- T')' 

Z Z'  z  —  z 

for  the  equation  of  conical  surfaces  in  general,  the  function  tp  depend- 
ing entirely  on  the  directrix. 

(136.)  Eliminating  the  function  (p,  by  differentiating  each  member 
of  this  equation  with  respect  to  x  and  y,  and  dividing  the  results  as  in 
(58),  we  have 

{y  —  y')  p'  _  s  —  z'  —  {.X  —  x')  p' 

z  —  z'  —  (y  —  y')  cl  (a?  —  a;')  g' 

which  reduces  to 

z  —  z'  =  p'  {X  —  x')  +  9'  (J/  —  ^0, 
the  differential  equation  of  conical  surfaces  in  general. 

(137.)  This  same  equation,  like  that  of  cylindrical  surfaces,  may 
be  obtained  more  readily  by  the  consideration  of  the  tangent  pleme, 
which,  as  it  always  passes  through  the  vertex  (x',  y',  z',)  is,  in  every 
position,  represented  by  the  equation 

z  —  z'  =  p'  {x  —  x')  +  q'  (y  —  y'\ 
this  relation,  therefore,  must  exist  between  the  partial  differential  co- 
efficients p',  q',  for  every  point  of  the  surface,  in  order  that  it  may  be 
conical. 

As  in  Problem  I.  if  for  p'l  q',  we  substitute  their  values  derived 
from  the  implicit  equation  of  any  conical  surface,  the  differential  equa- 
tion becomes 

(x  —  x')  —  -{-  (y  —  y')  —  4-  iz  —  z')  -r  =  Q- 
ax  ay  dz 

PROBLEM  IV. 

(138.)  Given  the  position  of  the  vertex,  to  determine  the  equation 
of  the  conical  surface  that  envelopes  a  given  curve  surface. 

Since  the  cone  envelopes  the  proposed  surface,  the  curve  of  con- 
tact is  common  to  both,  so  that  the  tangent  planes  to  the  cone  touch 
also  the  given  surface,  according  to  this  curve.  The  equation,  there- 
fore, of  the  tangent  plane 

z~z'  =  p'{x  —  x')  +  q'(y  —  y')  ....  (1); 


172  THE  DIFFERENTIAL  CALCULUS. 

holds  equally  for  any  point  on  the  conical  surface,  and  for  any  point 
in  the  curve  of  contact.  Hence,  if  the  values  of  p',  q',  be  derived 
from  the  equation  of  the  given  surface,  and  substituted  in  (l),  this, 
combined  with  the  equation  of  the  given  surface,  must  represent  the 
curve  common  to  both  surfaces,  that  is,  the  directrix  of  the  cone. 
Therefore,  the  vertex  and  directrix  being  known,  the  equation  of  the 
required  conical  surface  becomes  determinable. 

(139.)  If  the  given  curve  surface  be  of  the  second  order,  the  equa- 
tion (1)  will  be  also  of  the  second  order ;  but,  nevertheless,  the  com- 
bination of  these  two  equations  will  be  that  of  a  plane,  for  a  surface 
of  the  second  order  may  be  generally  represented  by  the  equation 

Ar'+Bf     -t-  Cz^     ) 
-f-  2Dyz  +  2F.XZ  +  2Fxy  V  =  K  .  .  .  .  (2), 
+  2Gtj    -\-2Urj    +2Js     j 

which  gives 

d2  ___  Aa^  +  Fy  +  E2  4-  G- 

dx  F.x  +  By  +  Cz  +  J 

de_        Fx  +  By  +  Dz  +  H 

d^  Ex  +  Dy  +  Cz  +  J  ' 

substituting  these  values  for  p'  and  q',  in  the  equation  (1),  and  sub- 
tracting from  the  result  the  equation  (2),  we  have. 


{Ax'  +  Fy'  -\-  Es'  +  G)x 
-t-  {Fx  +  By'  +  Dz'  +  H)  y 

+  {Ex'  +  By'  +  Cz'  +  J)   z 
+    Gx'  +  Uy'  +  Jz'  +  K 


=  0  .  .  .  .  (3), 


which  is  the  equation  of  a  plane  ;  therefore,  the  conical  surface  which 
circumscribes  a  surface  of  the  second  order,  must  itself  be  also  of  the 
second  order. 

(140.)  The  above  proof  is  from  JWonge  {Application  de  P Analyse 
a  la  Geome/rzc),  but  it  may  be  rendered  much  more  concise,  by  assu- 
ming the  axes  of  reference  so  as  to  give  the  general  equation  of  the 
surface  a  simpler  form.  Thus,  let  the  axes  of  x  pass  through  the 
centre,  if  the  surface  have  a  centre,  or  be  parallel  to  its  diameters  if 
it  have  not,  and  let  the  other  two  axes  be  parallel  to  the  conjugates 
to  this,  the  form  of  the  equation  will  then  be 

Ar*  +  Bi/^  +  Ca;2  +  2Fz  =  G (4), 


THE  DIFFERENTIAL  CALCULUg.  173 

dz  _  _  F  +  Cx   dz  __        By 

'  '  dx  Az     ^  dy  Az' 

These  values,  substituted  for  p'  and  g'  in  (1),  convert  that  equation 

into 

.       .'  4.  {F  +  Cx){x-x')        By{y-y')  _ 
z-z   + +  ^^  0, 

or 

As^  +  By^  +  Cx^  +  Fx  —  Az'z  —  B^'y  —  Gxlx  —  Fx'  =  0. 
The  difference  between  this,  and  (4),  is 

Az'z  +  By'y  +  Cx'x  +  Fx  +  Fx'  =  G, 
the  equation  of  a  plane. 

(141.)  Referring  again  to  Mongers  process,  we  may  remark,  that 
if  we  accent  the  constants  in  the  general  equation  (2),  it  maybe  taken 
as  the  representative  of  another  surface  of  the  second  order,  for  which 
the  plane  of  contact  with  a  circumscribing  cone,  whose  summit  coin- 
cides with  that  of  the  former  cone,  will  be  represented  by  the  equa- 
tion 

(Ax'  +  Fy'  +  Es'  +  G')  x'\ 
+  {Fx  +  By'  +  Bz'  +  U')yf  , 

+  (Ex'  +  Dj/'  +  Cz'  +  J')  «(  ~  "  •  •  •  •  ^^^• 
+   Gx'  +  By  +  Js'  +  K')    ) 

Now,  although  equation  (3)  be  multiplied  by  an  indeterminate  con- 
stant, p,  the  result  will  still  represent  the  same  plane,  and  this  plane 
will  obviously  be  identical  to  that  represented  by  (5),  provided  the 
coefficients  of  the  variables  x,  y,  z,  are  the  same  in  both  equations, 
that  is  to  say,  provided  we  have  the  conditions 

p  {Ax'  +  Fy'  +  Fz'  +  G)  =  Ax'  +  Fy'  +  Fz'  +  G' 
p  (Fx'  +  By'  +  Dz  +  H)  =  Fx'  +  By'  +  Dz'  +  H' 
p  {Fx'  +  By'  +  Cz'  +  J)  =  Ex'  +  By  +  Cz'  +  J' 
p  (Gx'  +  Hy'  +  Jz'  +  K)  =  Gx'  +  W  +  Jzf  +  K'. 

As,  therefore,  the  four  quantities  x',  y',  z',  p,  are  arbitrary  they 
may  be  determined  so  that  these  conditions  shall  be  fulfilled,  the  four 
equations  being  just  sufficient  to  fix  the  values  of  these  four  quanti- 
ties, and  as  each  of  them  enters  only  in  the  first  degree,  they  will 
each  have  but  one  value.  It  follows,  therefore,  that  there  is  a  certain 
point,  and  only  one,  from  which,  as  a  vertex,  if  tangent  cones  be 


174  THE  DIFFERENTIAL  CALCULUS. 

drawn  to  two  given  surfaces  of  the  second  order,  their  planes  of  con- 
tact shall  coincide  The  common  vertex  will  be  at  the  intersection 
of  those  diameters  to  each  of  which  the  plane  of  contact  is  conjugate ; 
since  it  has  been  shown  above,  that  the  vertex  of  the  tangent  cone  is 
always  situated  on  that  diameter  of  the  surface,  to  which  the  plane  of 
contact  is  conjugate.* 

(142.)  We  may  here  observe,  that  as  we  have  not  fixed  the  origin 
of  the  axes  to  any  particular  point  on  the  diameter  which  has  been 
taken  for  the  axis  of  x,  nor,  indeed,  the  diameter  itself,  we  may  con- 
sider the  diameter  to  be  that  passing  through  the  vertex  {x',  y\  2',)  of 
the  cone,  and  this  point  to  be  the  origin,  in  which  case  x',  ?/',  z',  will 
each  be  0,  and  the  equation  of  the  plane  through  the  curve  of  contact, 
will  then  be  simply 

Fx  =  G.'.x=^, 

hence,  the  plane  through  the  curve  of  contact,  is  conjugate  to  the 
diameter  through  the  vertex  of  the  cone.  If  this  vertex  be  supposed 
infinitely  distant,  the  same  result  will  belong  to  the  circumscribing 
cylinder,  viz.  that  the  plane  of  the  curve  of  contact,  is  conjugate  to 
the  diameter  parallel  to  the  generatrix  of  the  cylinder. 

Surfaces  of  Revolution. 

(143.)  The  surfaces  of  revolution,  considered  in  the  Analytical 
Geometry,  comprise  those  only  in  which  the  revolving  curve  is  always 
situated  in  the  plane  of  the  fixed  axis.  We  shall  here  treat  of  sur- 
faces of  revolution  in  general,  the  revolving  curve  being  any  how 
situated  with  respect  to  the  axes.  Sections  of  the  surface,  in  the 
plane  of  the  axis,  are  called  meridians. 

PROBLEM    V. 

(144.)  To  determine  the  equation  of  surfaces  of  revolution  in 
general. 

Let  the  equations  of  the  generating  curve  be 

F{x,y,z)=^Oj{x,y,z)  =  0  ....  (1), 

*  For  these,  and  other  kindred  properties,  the  student  is  referred  to  Mr,  Davits'' t 
paper  on  Geometry  of  Three  Dimensions,  in  LeyhourrCs  Repository,  vol.  5. 


THE   DIFFERENTIAL  CALCULUS.  175 

and  those  of  the  fixed  axis 

then,  since  the  characteristic  property  of  surfaces  of  revolution  is, 
that  every  section  perpendicular  to  the  fixed  axis  is  a  circle,  we  shall 
have  first  to  determine  a  plane  perpendicular  to  the  line  (2),  and  then 
to  express  the  condition  that  this  plane,  combined  with  the  surface, 
always  represents  a  circle  whose  centre  is  on  (2).  Now  the  equa- 
tion of  the  required  plane  is  {Anal.  Geom.) 

z  -\-  ax  -^  by  =  c  .  .  .  .  (3), 
and  the  condition  is,  that  it  must  give  the  same  section  as  if  it  were  to 
cut  a  sphere,  whose  centre  we  may  fix  at  pleasure,  but  whose  radius 
will  vary  with  the  section,  that  is,  it  will  depend  upon  c  in  equa.  (3). 
Assuming  the  centre  of  this  sphere  at  the  point  where  the  line  (2) 
pierces  the  plane  of  xy,  its  equation  will  be  {Anal.  Geom.) 
{x—  aY+  {y  —  l3f  +  z'^r'  .  .  .  .  (4). 
Hence,  supposing  r  to  be  the  proper  function  of  c,  the  equations  (1), 
(3),  (4),  must  all  have  place  together;  hence  we  may  eliminate  x, 
y,  s,  and  thus  determine  what  the  relation  between  r  and  c  must  ne- 
cessarily be,  to  render  these  equations  coexistent.     The  result  of  the 
eUmination  will  obviously  lead  to  c  =  (pr',  hence,  substituting  for  c 
and  r  their  values  in  terms  of  the  variables,  we  have,  finally, 

z  +  ax+  by  =  cp  \{x  —  ay  +  {y—  (3f  +  z"^   ....  (6), 
forthe  relation  which  must  always  exist  among  the  coordinates  of  every 
point,  in  every  circular  section.     This,  therefore,  is  the  equation  of 
surfaces  of  revolution  in  general. 

(145.)  If  the  fixed  axis  be  taken  for  the  axis  of  2,  then  a,  a;  6, /3, 
are  each  0,  therefore,  in  this  case,  the  general  equation  becomes 

z=<p{x'  +  f-{-2?)  .  .  .  .  (6), 
which,  solved  for  z,  takes  the  form 

2  =  4;(x^  +  y^) (7). 

(146.)  There  is  one  case  of  this  general  problem,  viz.  that  where 
the  generatrix  is  a  straight  line,  revolving  round  the  axis  of  z,  but  not 
in  the  same  plane  with  it,  that  deserves  particular  notice. 

Let  us  take,  for  axis  of  x,  the  shortest  distance  between  the  axis  of 


176  THE  DIFFERENTIAL  CALCULUS. 

2  and  the  generating  line ;  then  this  axis  will  be  perpendicular  to  botb 
(65),  the  equations,  therefore,  of  the  line  will  be 

X  ^  ±  a,  y  =  dc  bz, 
also,  for  any  variable  section  perpendicular  to  the  axis  of  2 

2  =  c,  a^  +  y^  +  2^  =  »^. 
Eliminating  x,  y,  2,  we  have 

a^  +  6-c2  +  c''  =  1^, 
for  c  and  1^  putting  their  values  above,  we  have 

a*  +  6^0^  =  x'-^yK 
By  putting  successively  x  =  0,  x  =  0  in  this  equation,  the  result- 
ing forms  belong  to  hyperbolas,  hence  the  surface  is  the  hyperboloid 
of  revolution  of  a  single  sheet.     The  equation  of  the  hyperbola  cor- 
responding to  a?  =  0  is 


so  that  y  =  ±  bz  is  the  equation  of  the  asymptotes,  (see  Anal.  Geom.) 
hence  the  generating  straight  line,  in  its  first  position,  is  in  a  plane 
with  and  parallel  to  one  or  other  of  the  asymptotes  of  that  hyperbola 
in  its  first  position,  which  would  generate  by  revolving  round  the  axis 
of  2,  the  same  surface  as  the  line ;  these  two  lines,  therefore,  continue 
parallel  during  the  revolution  of  both ;  the  one,  viz.  the  asymptote, 
generating  the  conical  surface  asymptotic  to  the  hyperboloid  genera- 
ted by  the  other  line,  viz.  the  line 

X  =  ±  a,  y  =  bz, 

or 

X  =  ±  o,,y  =  —  6z, 

and  it  therefore  follows  that  these  four  lines  will  be  the  sections  made 
on  the  surface  by  two  tangent  planes  to  the  asymptotic  cone  drawn 
through  any  diametrically  opposite  points  in  its  surface  ;  these  will 
cut  each  other  on  the  surface  two  and  two,  and  include  an  angle 
equal  to  that  between  the  asymptotes,  so  that  the  surface  may  be 
generated  by  the  revolution  of  either  of  these  intersecting  lines. 

We  shall  shortly  see  that  hyperboloids  of  one  sheet,  in  general, 
admit  of  two  distinct  modes  of  generation  by  the  motion  of  a  straight 
line. 

( 147. )  EUminating  the  indeterminate  function  9,  which  depends  on 


THE  DIFFERENTIAL  CALCULUS.  177 

the  nature  of  the  generating  curve  (1)  by  differentiation,  as,  in  the 
preceding  problems,  we  find 

p'  -\-  a  __  X  —  a+  p'z 

q'  -\-  b       y  —  (3  +  q'z^ 
from  which  results  the  partial  differential  equation 
(^y_l3^bz)p'—{x—a—az)q'—b{x—a)—a{y—(3)  =  0.  .  (1), 

and  when  the  axis  of  z  coincides  with  that  of  revolution,  this  becomes 
j/p'  —  xq'  =  0. 
The  differential  equation  of  surfaces  of  revolution  may  also  be  ob- 
tained from  the  consideration  of  the  normal,  which  must  always  cut 
the  axis  of  revolution,  being  situated  in  the  meridian  plane.  Thus 
the  equations  to  the  normal  are  (125) 

x  —  x'  +  p'{z  —  z')  =  0  ) 

y-y'  +  q'{z-z')  =  oi 
and  as  these  must  exist  simultaneously  with  the  equations  (2),  we 
may  eliminate  x,  y,  z,  and  the  result  will  necessarily  be  the  required 
relation  between  p',  q,  and  the  variable  coordinates  x,  y',  z',  of  any 
point  on  the  surface. 

PROBLEM    VI. 

(148.)  A  given  curve  surface  revolves  round  a  given  axis,  to  de- 
termine the  surface  which  touches  and  envelopes  the  moveable  surface 
in  every  position. 

The  enveloping  surface  touches  the  moveable  one  in  every  posi- 
tion ;  if,  therefore,  we  take  any  particular  position  of  the  latter,  their 
combination  will  give  the  curve  of  contact ;  this  curve  being  common 
to  both  surfaces,  the  tangent  planes,  at  all  its  points,  are  common  to 
both  surfaces;  hence,  the  values  ofp',  q\  which  vary  only  with  the 
tangent  plane,  are  the  same  for  both  surfaces,  as  far  as  this  common 
curve  is  concerned,  and  it  is  evidently  by  the  revolution  of  this  cui-ve 
round  the  fixed  axis,  that  the  enveloping  surface  is  generated.  Hence, 
to  determine  this  curve,  we  must  deduce  p,  q\  from  the  given  equa- 
tion, substitute  them  in  the  general  equation  (1)  of  surfaces  of  revolu- 
tion, since  there  is  a  line  on  some  such  surface  to  which  they  belong, 
as  well  as  to  the  given  surface ;  and  then,  to  determine  what  this 
line  really  is,  it  will  be  necessary  merely  to  combine  this  last  result 

23 


178  THE  DIFFERENTIAL  CALCULUS. 

with  the  equation  of  the  given  surface :  we  shall  thus  obtain  the 
equations  of  the  generating  curve,  and  the  position  of  the  fixed  axis 
being  previously  known,  the  enveloping  surface  is  determinable  by 
Prob.  V. 

(149.)  As  an  illustration  of  this,  let  us  suppose  a  spheroid  to  re- 
volve about  any  diameter,  to  find  the  equation  of  the  surface  envelop- 
ing it  in  every  position. 

Let  the  siu-face  be  referred  to  the  principal  diameters  of  the  sphe- 
roid, then  the  equations  of  any  other  diameter  will  be 

X  =  az,ri  =  bz  .  .  .  .  (1), 
and  the  spheroid  itself  may  be  represented  by  the  equation 

^  +  2/'  ■+■  n'z^  =  vp?, 
from  which  we  derive 

1     ^     ,  1     y 

'^  vir     z    ^  n"     z 

substituting  these  values  in  the  general  equation,  for  all  surfaces  of 
revolution  round  the  proposed  axis  (1),  that  is  in  the  equation 

{y  —  bz)  p'  —  {x  —  az)  q'  +  ay  —  6a;  =  0, 
and  we  have 

{ay  —  bx)  (1  —  -1)  =  0, 

.'.  ay  =  bx, 
hence,  combining  this  with  the  given  equation,  we  have,  for  the  gene- 
rating curve  of  the  envelope,  the  equations 

or'  +  2/^  +  »V  =  mM  ,^. 

ay  =  bx  I    •  •  •  •   ^  /' 

hence,  the  envelope  itself  is  to  be  determined  thus.  We  must  eUmi- 
nate  x,  y,  z,  by  means  of  (2),  and  the  equations 

z  -\-  ax  -\-  by  =  c   \  ,„. 

of  any  circular  section,  the  result  will  be 


(^„a  _  m')  {a"  +  b-)  =  {c  Vn'  —  1  —  Vm'  —  ry, 
putting  for  r  and  c  their  values  in  terms  ofx,  y,  z,  we  have  finally, 
{iv'  {r"  +  f -{-  z")  —  ni'l  {a  +  6^), 
=  \{z  +  ax-\-  by)y/n^—l  —  Vm^  —  x'—f  —  :^l^ 
which  is  the  equation  of  the  enveloping  surface.* 

*  This  solution  is  from  Hymer's  Geometry  of  Three  Dimensims,  p.  145. 


I 


THE  DIFFERENTIAL  CALCULUS.  179 


CHAPTSR    III. 
ON  THE  CURVATURE  OF  SURFACES  IN  GENERAL. 

(160.)  The  simplest  method  of  contemplating  surfaces,  is  by  con- 
sidering them  as  produced  by  the  motion  of  a  line  straight  or  curved, 
which,  in  all  its  positions,  is  subject  to  a  fixed  law.  Viewed  under 
this  aspect,  surfaces  seem  to  divide  themselves  into  two  distinct  and 
very  comprehensive  classes,  viz.  those  whose  generatrices  must  ne- 
cessarily be  curves,  and  those  whose  generatrices  may  be  a  straight 
line.  If,  in  this  latter  class  of  surfaces,  the  law  which  regulates  the 
generating  straight  line  be  such  that  through  any  two  of  its  positions, 
however  close,  a  plane  may  always  be  drawn,  then  it  is  obvious,  that 
in  every  such  surface,  if  a  plane  through  the  generatrix  in  any  posi- 
tion, but  not  through  any  other  points  of  the  surface,  that  is  if  a  tan- 
gent plane,  be  drawn,  this  plane,  if  supposed  perfectly  flexible,  might 
be  wrapped  round  the  surface,  without  being  twisted  or  torn,  or,  on 
the  contrary,  the  surface  itself  might  be  unrolled,  and  would  then  co- 
incide in  all  its  points  with  the  plane.  Surfaces  of  this  kind  are,  there- 
fore, very  properly  distinguished  by  the  name  Developable  Surfaces ; 
the  simplest  of  these  are  the  cone  and  cylinder. 

(151.)  We  see,  therefore,  that  these  surfaces  are  such  that  a  plane 
may  be  drawn  through  any  two  positions  of  the  generatrix,  and  which 
if  turned  round  one  position  supposed  fixed,  will  pass  through  all  the 
intermediate  positions  of  the  other.  But  if  the  law  of  generation  is 
such  that  this  cannot  have  place  for  any  two  positions,  however  close, 
then  the  tangent  plane,  through  one  position,  could  plainly  never  be 
brought  to  pass  also  through  another  position,  however  near,  without 
being  twisted.  Such  surfaces,  therefore,  are  properly  designated  by 
the  name  Tioisted  Surfaces. 

These  two  kinds  of  surfaces  will  be  separately  discussed  hereafter, 
the  particulars  in  the  present  chapter  relate  to  curve  surfaces  in  gene- 
ral. 

Osculation  of  Curve  Surfaces. 

(152.  Let  the  equations  of  two  curve  surfaces  be 


180  THE  DIFFERENTIAL  CALCULUS. 

when  referred  to  the  same  axes  of  coordinates.  The  first  of  these 
surfaces  we  shall  suppose  fixed,  both  in  magnitude  and  position  by  the 
constants  a,  b,  c,  &c.,  which  enter  its  equation,  being  fixed.  The 
second  surface  we  shall  suppose  fixed  only  in  form,  by  the  form  of  its 
equation  being  given,  but  indeterminate  as  to  magnitude  and  position, 
on  account  of  the  arbitrary  constants,  A,  B,  C,  &c.,  which  enter  its 
equation. 

Let  now  the  variables  x,  and  y,  take  the  increments  h  and  k,  then, 
for  the  first  surface,  we  have  (60) 

,   dz  ,     ,    dz  .    .    ^    ,d^z  .„    ,    ^   d^2    ,,    , 
z'  =  z  +  —  h-{-^k  +  ^  (-—  r-  +  2  -j—^  hk  + 
ax  ay  "   ^rfar  dxdy 

and  for  the  second, 

Z'  =  Z  +  —  /i  +  —  A;  +  X  (f??  +2  —  hk  + 
dx  dy  ^      dx'  dxdy 

or,  more  briefly, 

z'  =  z-\-  p'h  +  q'k  +  i  (r'K"   +  2s'hk  +    t'¥)  +  &c. 
Z  =  Z+  P7i  +Q'fc  +  1  (R'h'  +  2S'hk  +  Tkr')  +  &c. 

Now  the  constants  A,  B,  C,  &c.  being  arbitrary,  we  may  determine 
one  of  them  in  functions  of  x,  y  and  the  known  constants,  so  that  the 
condition 

z=  Z  or f{x,y)  =F  {x,y) 
may  be  fulfilled.  Such  a  value  substituted  for  the  constant  in  the 
equation  Z  =  F  (x,  y,)  will  cause  all  the  surfaces  represented  by 
this  equation  to  have  a  point  {x,  y,  z,)  in  common  with  the  given  sur- 
face. If  two  more  of  the  arbitrary  constants  be  determined  from  the 
conditions 

p'  =  P',  q  =  Q', 
the  resulting  values  of  these  constants  being  also  substituted  in  the 
same  equation,  the  surfaces  then  represented  will,  in  consequence, 
all  have  a  common  tangent  plane  at  the  point  {x,  y,  z,)  with  the  fixed 


THE  DIFFERENTIAL  CALCULUS.  181 

surface.  Therefore,  that  this  may  be  the  case,  three  arbitrary  con- 
stants, at  least,  must  enter  the  proposed  equation,  and  the  contact 
which  they  determine  is  called  contact  of  the  first  order.  Contact  of 
the  second  order  requires  that  the  foUowuig  additional  conditions  be 
fulfilled,  viz, 

,.'  =  R',  s'  =  S',  t'  =  T', 

requirmg  three  more  arbitrary  constants  to  be  determined,  and  so  on ; 
and  that  surface,  all  whose  arbitrary  constants  are  determined  agreea- 
bly to  these  conditions,  will,  for  reasons  similar  to  those  assigned  at 
(87)  for  plane  curves,  touch  the  proposed  surface  more  intimately 
than  any  other  surface  of  the  same  order.  It  is  called  the  osculating 
surface  of  that  order. 

If  the  touching  surface  be  a  sphere,  then,  since  in  its  equation  there 
can  enter  only  four  disposable  constants,  the  contact  cannot  be  so  high 
as  the  second  order,  seeing  that  for  this  there  must  be  six  disposable 
constants,  but  as  contact  of  the  first  order  would  leave  still  one  con- 
stant arbitrary,  it  follows  that  an  infinite  number  of  spheres  may  have 
simple  contact  with  a  surface  at  any  proposed  point,  yet  one  of  these 
may  be  determined  that  shall  be  strictly  the  osculating  sphere,  or 
which  shall  touch  more  intimately  all  round  the  point  of  contact  than 
any  other. 

Curvature  of  different  Sections. 

PROBLEM  I. 

(163.)  At  any  point  on  a  curve  surface  to  find  the  radius  of  cur- 
vature of  a  normal  section. 

For  greater  simplicity,  let  us  suppose  the  plane  of  xy  to  coincide 
with  the  tangent  plane  at  the  proposed  point,  then  the  axis  of  z  will 
coincide  with  the  normal,  and  all  the  normal  sections  will  be  vertical. 
Let  the  plane  of  the  proposed  section  be  incUned  at  an  angle  6  to  the 
plane  ofxz,  then  the  angle  which  its  trace  x'  on  the  plane  ofxy  makes 
with  the  axis  of  x  will  obviously  be  6,  and  the  x,  y  of  this  trace  will 
also  be  the  x,  y  of  the  section.     Now  (97)  the  radius  of  curvature  p 

at  the  proposed  point  where,  (86),  (dr')  —  (tfe)  and  j;r-7z  =  0,  is 


188  THE  DIFFERENTIAL  CALCULUS. 


d^ 

rf«« 

dor" 

dx' 

^       d^z 
dx' 

dx           dx' 

But  (86) 

dx''  _  ds'  _ 
dx"       dx" 

1  +    j^  =  1  +  tan."  6 ; 
dx' 

hence,  by  substitution, 

1  +  tan.«  6              _ 

1 

V  +  2s  tan.  e  +  «'  ten.*0  r'  cos.*  6  +  2s  cos.  6  sin.  6  +  t'  sin.«  fl  * '  ^  *' 
For  the  radius  of  curvature  p',  of  a  second  normal  section  inclined  at 
an  angle  6  +  90°  to  the  plane  of  xz,  we  have,  by  putting  6  +  90** 
for^, 

1 

in.^  6  —  2s'  COS. 
consequently, 


f*        r'  sin."  6  —  2s'  cos.  6  sin.  6  +  V  cos."  &    '  '  '  '  ^^^' 


P         P 
so  that  the  sum  of  the  curvatures  of  any  hoo  normal  sections  through 
the  same  point  at  right  angles  to  each  other,  is  c  constant  quantity. 

(154.)  Consequently,  when  one  of  these  curvatures  is  the  greatest 
possible,  the  other  must  be  the  least  possible ;  that  is,  at  every  point 
on  a  curve  surface,  the  sections  of  greatest  and  least  curvature  are  al- 
ways perpendicular  to  each  other,  which  beautiful  theorem  was  first 
discovered  by  Euler,  and  is  demonstrated  by  most  writers  on  curve 
surfaces,  though  in  a  manner  far  less  simple  than  that  above. 

(156.)  To  determine  the  values  of  the  radii  of  curvature  of  any 
perpendicular  sections  at  their  point  of  intersection,  let  the  plane  of 
xz  be  made  to  coincide  with  one  of  them  by  turning  round  the  nor- 
mal ;  that  is  to  say,  let  6  =  0,  then  the  foregoing  expressions  for  p 
and  p'  become 

(156.)  But  to  determine  the  expressions  for  the  radii  of  greatest 
and  least  curvatures,  without  causing  the  vertical  planes  of  coordinates 
to  coincide  with  the  sections,  we  must  know  the  inclinations  of  these 


THE  DIFFERENTIAL  CALCULUS.  183 

sections  to  the  vertical  planes,  that  is,  we  must  know  the  angle  6.  To 
find  this  from  the  property  —  =  max.  or  min.  we  have,  taking  6  for 
the  independent  variable  in  the  expression  for  p, 

-_£.  =  —  2r'  cos.  6  sin.  d  +  2s'  (cos.^  6  —  sin."  6)  + 

0)0 

2t'  sin.  6  cos.  6  =  0....  (5), 
or,  dividing  by  2  sin.^^,  we  have 

cot."  6  +  *-^  cot.  d  _  1  =  0 (6), 

s 

from  which  we  get  for  the  two  inclinations  sought 

r'  —  t'  ±  V  (r'  —  t'Y  +  ^      cos.  6 
cot.  ^  = ~ 


2s'  sin.  6 

the  upper  sign  corresponding  to  the  meiximum,  and  the  under  to  the 
minimum.  Substituting  these  values  in  the  first  of  the  expressions 
(1),  which  may  be  written  thus 

_  cot."  6  +  1 

^  ~  r'  cot."  6  +  2s'  cot.  6  +  f' 
we  have  for  the  radii  of  greatest  and  least  curvatures  the  expressions 

2 


r'-\-t'-\-V{r'  —  t'r-^As"(  _ 

2  >    '  *  '  ^  ■'* 

R= —  =C 

These  are  called  the  principal  radii  of  curvature  at  the  proposed 
point,  and  the  sections  themselves  the  principal  sections  through  that 
point. 

(167.)  If  we  know  the  principal  radii  and  the  inclination  <p  of  any 
normal  section  to  a  principal  section  through  the  point,  the  radius  of 
curvature  of  the  normal  section  at  that  point  may  be  expressed  in 
terms  of  these  known  quantities.  For,  bringing  the  vertical  coordi- 
nate planes  mto  coincidence  with  the  planes  of  principal  section,  we 
have  6  =  0,  and,  consequently,  as  appears  from  equation  (6),  last 
article,  »'  =  0 ;  and,  since  (4) 

^-=r'^  =  t' 


184  THE  DIFFERENTIAL  CALCULUS. 

we  have 

r'  cos.^  cp  -^  t'  sin.^  9       r  sin.^  9  +  R  cos.29 

•••  -  =  B-^—  +  ^- (9)- 

p        K  sm.  9        r  COS.  9 

It  is  plain  from  this  expression  that  if  R  and  r  have  the  same  sign, 
p  will  have  that  sign  for  every  section  through  the  proposed  point, 
which  is  the  same  as  saying  that  if  the  principal  sections  are  both 
convex  or  both  concave,  every  other  section  through  the  same  point 
will  be  similarly  convex  or  concave,  and,  therefore,  also  the  entire 
surface  at  that  point.  In  such  a  case  the  minimum  radius  must  be 
absolutely  shorter  than  any  other  radius  of  curvature  at  the  point,  and 
the  maximum  radius  longer  than  any  other.  , 

(158.)  If  the  two  prmcipal  radii  have  not  only  the  same  sign  but 
the  same  length,  then  the  foregoing  expression  gives  always  p  =  R 
whatever  be  the  inclination  9,  so  that  then  all  the  normal  sections 
have  the  same  curvature  and  all  are  principal  sectioiis,  as  is  the  case 
with  the  sphere  and  with  the  ellipsoid  of  revolution,  the  paraboloid  of 
revolution,  &c.  at  those  points  through  which  the  fixed  axis  passes. 

(159.)  If  the  surface  belong  to  the  second  of  the  classes  mention- 
ed in  (147),  then  no  point  can  be  assumed  on  it  through  which  a 
straight  line  may  not  be  drawn,  and,  as  the  curvature  of  this  line  is  0, 
it  follows  that  the  curvature  of  the  section  perpendicular  to  it  must 
be  equal  to  the  sum  of  the  curvatures  of  any  two  perpendicular  sec- 
tions through  the  same  point. 

(160.)  Let  us  now  suppose  that  the  principal  radii  R,  r  have  dif- 
ferent signs,  as  r  positive  and  R  negative,  which  will  be  the  case  if 
one  of  these  sections  be  convex  and  the  other  concave,  we  shall  then 
have 

_  Rr 

R  COS."  9  —  r  sin.^  9' 
which  becomes  infinite  when 

R 

r  sin.^  9  =  R  cos.^  9  or  when  tan.  9  =  ±  %/  — , 

r 

but  for  all  positive  and  negative  values  of  9  between  this  and  0,  p  will 
be  positive,  while  beyond  these  limits  p  will  be  negative. 

It  appears,  therefore,  that  if  from  the  origin  two  straight  lines  be 


THE  DIPFEHENTIAL  CALCULUS.  i  85 

drawn  in  the  tangent  plane  inclined  to  the  axis  of  x  at  the  angles  9  = 

R  R 

+  \/  —  and  ffl  =  —  v^  — ,  these  will  coincide  with  the  surface ;  all 
r  r 

the  sections  between  the  sides  of  the  two  opposite  angles  thus  form- 
ed will  be  convex,  all  the  sections  between  the  sides  of  the  other  two 
opposite  supplementary  angles  will  be  concave,  so  that  the  two  straight 
lines  which  we  have  seen  may  be  drawn  from  the  proposed  point  to 
coincide  with  the  surface,  separate  the  convexity  from  the  concavity 
at  that  point. 

(161.)  In  order  to  determine  whether  the  principal  radii  at  any  point 
are  both  of  the  same  sign  or  not,  we  may  observe  that  the  expressions 
(7)  for  these  radii  at  art.  (166)  may  be  put  under  the  form 

2 


r  +(  -\-  v/  (r'  -f  if  —  4  {r'(  —  s'^)  ) 
R  ^ 2  V  .  .  .  .  (10), 

r'  +  <'  —  v/  (r'  +t'f  —  4.{7f—  s'-)  ) 
from  which  forms  we  immediately  see  that  the  radii  will  have  the  same 
sign,  viz.  positive  if  r'/'  —  s'^  >  0,  and  contrary  signs  if  r'i'  —  s'^  < 
0 ;  this  last  condition,  therefore,  exists  in  the  case  just  considered. 

(162.)  We  shall  terminate  these  remarks  by  showing  that  a  para- 
boloid of  the  second  order  may  always  be  found,  such  that  its  vertex 
being  applied  to  any  point  in  any  curve  surface,  the  normal  sections 
through  that  point  shall  have  the  same  curvature  for  both  surfaces. 

For,  take  the  planes  of  the  principal  sections  for  those  of  ar,  yz, 
then  the  radii  of  these  sections  being  R,  r  we  know  that  a  paraboloid, 
whose  vertex  is  at  the  origin,  will  in  reference  to  the  same  axes  be 
represented  by  the  equation  {Anal.  Geom.) 

r  and  R  being  the  semi-parameters  of  the  sections  of  the  paraboloid 
on  the  planes  of  xz,  yz.  Now  the  equation  of  a  normal  section  of 
this  paraboloid,  by  a  plane  whose  inclination  to  that  of  X2  is  <p,  will  be 
obtained  by  substituting  in  this  equation  x'  cos.  (p  for  x,  x'  sin.  <p  for  y, 
z  remaining  the  same  for  all  normal  sections  {Anal.  Geom.) ;  hence, 
the  equation  of  the  section  in  question  is 

__    CO8.-9       sin.^  (p      ,j   ^     ,,  __  2Rr 

^       ^'  2r  2R^*    ■*''^'       R  cos.^  (p  ±  r  sin.'' (p  ^' 

24 


186  THE  DIFFERENTIAL  CALCULUS. 

SO  that  the  semi-parameter,  and,  consequently,  the  radius  of  curvature 
(94)  of  this  parabolic  or  hyperbolic  section,  is 

Rr 

R  cos.^  (p  db  r  sin.^  9' 
the  very  same  as  the  radius  of  curvature  of  the  corresponding  section 
of  the  proposed  surface,  be  this  what  it  may  (154).     Hence,  this  pa- 
raboloid has  the  same  curvature  in  every  direction  that  the  proposed 
surface  has  at  the  origin  of  the  coordinates. 

PROBLEM  II. 

(163.)  To  determine  the  radius  of  curvature  at  any  point  in  an  ob- 
hque  section. 

Take  the  tangent  to  the  section  through  the  point  as  axis  of  x,  the 

point  itself  for  the  origin,  and  the  axis  of  s  in  the  plane  of  the  section ; 

then,  calling  the  normal  the  axis  oi'z,  the  normal  section  throug-h  the 

axis  of  a;,  s,  and  the  oblique  section  a',  we  have,  at  the  proposed  point 

(ds'Y 
(86),  (ds)  =  {da').     Now  at  the  proposed  point  7  =       -  -,-  ,  but  if 

[a-z ) 

the  axis  of  z'  be  transferred  to  the  axis  of  Zf  then 

z  =  z'  COS.  6  .'.  {d^'z)  =  (d^z')  cos.  6 ; 
hence,  by  substitution, 

^  ^  Td^  ^^^'    ^  P  ^^^'  ^  •  •  •  •  (1)' 

where  7  is  the  radius  of  the  oblique  section,  and  p  the  radius  of  the 
normal  section  through  the  tangent  to  the  former ;  so  that  7  w  f  Ac  pro- 
jeciion  ofp  on  the  plane  of  the  oblique  section,  which  remarkable  pro- 
perty is  the  theorem  of  JMeusnier. 

It  immediately  follows  from  this  theorem,  that,  if  with  the  radius  of 
any  normal  section  of  a  curve  surface  a  sphere  be  described,  and 
through  the  tangent  to  that  section  at  the  normal  point  planes  be  drawn, 
cutting  both  the  sphere  and  the  proposed  surface,  every  section  of  the 
sphere  will  be  an  osculating  circle  to  the  corresponding  section  of  the 
surface,  because,  if  the  normal  radius  of  the  sphere  be  projected  on 
any  of  these  sections,  the  projection  will  obviously  be  the  radius  of 
,that  section,  and  the  same  projection  is,  by  the  above  theorem,  the 
I'adius  of  curvature  of  the  corresponding  section  of  the  proposed  sur- 


THE  DIFFERENTIAL  CALCULUS.  187 

Lines  of  Curvature  and  Radii  of  Spherical  Curvature. 

(164.)  In  speaking  of  plane  curves  we  have  already  explained  (104) 
what  is  to  be  understood  by  consecutive  normals  and  consecutive  curves. 
We  propose,  in  the  present  article,  to  consider  the  intersections  of 
any  normal  at  a  point  of  a  curve  surface  with  its  consecutive  normal ; 
but  here  it  must  be  remarked  that  consecutive  normals  to  curve  sur- 
faces do  not  necessainly  intersect,  as  in  plane  curves,  for,  before  coin- 
ciding, these  normals,  although  ever  so  close,  need  not  be  both  in  the 
same  plane ;  and,  in  such  a  case,  when  they  become  consecutive,  or 
coincide,  they  coincide  throughout  at  once,  having  even  then  no  point 
in  common  that  before  coinciding  was  a  point  of  intersection.  Hence 
such  consecutive  normals  have  no  point  of  intersection.  If,  however, 
upon  any  curve  surface  there  can  be  traced  a  line,  such  that  the  nor- 
mal to  the  surface  at  every  point  of  it  is  intersected  by  the  consecu- 
tive normal,  that  line  will  have  peculiar  properties.  Such  a  line  is 
called,  by  J\Ionge,  a  line  of  curvature. 

PROBLEM  in. 

(165.)  To  determine  the  lines  of  curvature  through  any  point  on  a 
curve  surface. 

Let  the  surface  be  referred  to  any  rectangular  axes  whatever,  then 
(.r',  i/,  z\)  being  any  point  on  it,  we  have,  for  the  equations  of  the 
normal, 

(A)  x  —  x'+p'{z  —  z')=Ol  n^ 

(B)  y  —  y'-\.q'{z  —  z')=0] (l)' 

Let  now  the  independent  variables  x',  y',  take  any  increments  h  k 
the  equations  of  the  normal  to  the  corresponding  point  will  be 
.     ,    dA   ,    ,    dA    ,    ,  \ 

ax  dy  J 

Now,  if  the  normals  (1),  (2)  intersect,  their  equations  must  exist 
simultaneously  ;  therefore,  since  A  =  0,  B  —  0, 
dA    ,   rfA    A-    ,  ^ 

dx         dy     k  \  i^. 

iB    ,    JB    Jr       ,  >  .  .  •  .  (A). 

dx"        dy'    h^  J 


188  THE  DIFPERENTIAL  CALCULUS. 

The  coordinates  {x,  y,  z,)  of  the  intersection  of  the  proposed  nonnals 
will  be  obtained  by  the  combination  of  the  four  equations  (1)  and  (3) 
in  terms  of  x\  y'  z',  which  are  fixed,  and  of  the  increments  k,  h. 
But  from  four  equations  three  unknowns  may  be  always  ehminated, 
and  the  result  of  this  elimination  will  be  an  equation  between  the 
other  quantities ;  hence  then  there  exists  a  constant  relation  between 
the  increments  k,  h,  when  the  normals  intersect,  these  increments  are 
therefore  dependent;  consequently  the  y,  x,  of  which  these  are  the 
increments,  must  be  dependent  ;*  therefore  when  the  normals  are 
consecutive,  that  is,  when  ^  =  0,  the  equations  (3)  become 
dA        dA      dy'    _ 


dx'        dy'  '  dx'  i  r„,v 

dB        dB_    4^ 

dxf        dy'  '  dx' 


:\- 


or,  by  substituting  for  A  and  B  their  values  (1), 

1  +  P'  ( P'  +  9'  ■^)  +  (='  -  ») Cr'  +  »■  -^j  =  0  .  .  .  .  (4), 

from  which,  eliminating  zf  —  2,  we  have  the  following  equation  for 

•  .      dy' 
determmmg  -^7 

((1  +  q'^)  S'  -p'q't')  ^  +  ((1  +  q")  r'-il+  r)t')  g-  - 

(1  +p'=)s'+p'gV  =  0      .  .  .   (6). 

This  being  a  quadratic  equation  furnishes  two  values  for  —  the 

tangent  of  the  incUnation  of  the  projection  of  the  line  of  curvature 
through  {x',  y',  2'),  on  the  plane  of  x?/  to  the  axis  of  x.  Hence,  there 
are  two  directions  in  which  lines  of  curvature  can  be  drawn  through 
any  proposed  point,  and  if  in  (6)  we  substitute  for  p',  q',  &c.  their 
general  values  in  functions  of  x,  y,  that  equation  will  then  be  the  dif- 
ferential equation  which  belongs  to  the  projections  of  every  pair  of 

*  If  this  should  appear  doubtful  to  the  student,  its  truth  may  be  shown  by  re- 
moving the  axes  of  a;,  y,  to  the  proposed  point,  in  which  position  k,  h,  will  be  the 
variable  coordinates  of  the  line  of  curvature,  and  these  will  merely  take  a  constant 
when  the  axes  are  replaced  in  their  first  position. 


THE  DIFFERENTIAL  CALCULrS.  189 

lines  of  curvature ;  so  that  every  line  on  a  curve  surface  which  at  all 
its  points  satisfies  this  equation,  will  be  a  line  of  curvature. 

(166.)  Between  every  pair  of  lines  of  curvature  there  exists  a  very 
remarkable  relation  :  it  is  that  they  are  always  at  right  angles  to  each 
other.  To  prove  this  it  will  only  be  necessary  to  place  the  coordi- 
nate planes,  which  have  hitherto  been  arbitrary,  so  that  the  plane  of 
xy  may  coincide  with,  or  at  least  be  parallel  to,  the  tangent  plane  at 
the  point  to  be  considered,  in  which  case  p'  and  q'  are  both  0,  and, 
consequently,  the  equation  (6)  becomes 

^  +  tzzl    ^_i^0         ..(7) 
do^  ^      s'      '  dx  "  •  •  •  •  UJ. 

therefore,  calling  the  two  roots  or  values  of -r^,  tan.  (p  and  tan.  9',  we 

have,  by  the  theory  of  equations, 

tan.  6  tan.  d'  =  —  1,* 
which  proves  that  the  projections  of  the  two  lines  of  curvature  through 
the  origin,  are  perpendicular  to  each  other,  and  consequently  the  lines 
themselves  are  perpendicular  to  each  other. 

Moreover,  the  equation  (7),  if  divided  by-^^  =  tan.^  5  becomes 

identical  to  equation  (6),  page  183,  which  determines  the  inclinations 
of  the  principal  sections ;  hence,  the  lines  of  curvature  through  any 
point,  always  touch  the  sections  of  greatest  and  least  curvature  at  that 
point.  Also,  in  the  same  hypothesis,  with  respect  to  the  disposition 
of  the  coordinate  planes  2'  =  0,  therefore  the  equation  (4)  or  (6) 
gives 

1  tan.  d 

r'  +  s  tan.  a  s'  +  /'  tan.  & 

but  if  the  plane  of  X2  coincide  Avith  a  plane  of  principal  section,  it 
will,  as  we  have  just  seen,  touch  the  line  of  curvature,  and  then  ^  =  0, 
so  that 

1  1 

r  t 

and  these  are  precisely  the  expressions  found  at  (152),  for  the  two 
radii  of  curvature  of  the  principal  sections  at  the  proposed  point,  in 

♦  Since  tangent  f  and  tangent  ^'  are  the  roots  of  equation,  (7),  and  —  1  is 
their  product,  recollecting  that  tangent  X  cot.  =  radius' =  1,  whence  0' is  the 
complement  of  <j).  Ed. 


190  THE  DIFFERENTIAL  CALCULUS. 

reference  to  the  same  axes  ;  hence  we  infer,  (161),  that  the  conse- 
cutive normals  to  the  surface  at  any  point,  intersect  at  the  same  points 
as  the  consecutive  normals  to  the  principal  sections.  These  points 
of  intersection,  are  no  other  than  the  centres  of  curvature  of  the  sur- 
face at  the  proposed  point,  for  if  spheres  be  described  from  these 
centres  to  pass  through  the  proposed  point,  they  will  touch  there, 
since  both  have  the  same  normal,  and  therefore  the  same  tangent 
plane ;  and  these  two  spheres  have  the  same  curvature  as  the  surface 
in  the  two  directions  of  the  lines  of  curvature,  since  consecutive  nor- 
mals to  the  surface  in  these  directions,  cut  that  through  the  point  at 
the  centres  of  these  spheres,  also  the  plane  sections,  tangential  to 
these  directions,  have  the  corresponding  sections  of  the  spheres  for 
their  osculating  circles,  since  the  consecutive  normals,  at  their  point 
of  contact,  also  intersect  at  these  centres ;  therefore,  the  radii  of  cur- 
vature of  the  surface  at  any  point,  coincides  entirely  with  the  radii  of 
curvature  of  the  principal  sections  through  that  point,  so  that  (156) 
if  the  radii  are  both  equal  at  any  point,  the  curvature  of  the  surface  is 
uniform  all  round  that  point. 

(167.)  The  annexed  figure  is  intended  to  give 
an  idea  of  the  disposition  of  the  lines  of  curvature 
on  the  surface  (S),  drawn  through  points  P,  P',  &c. 
PT,  P'T',  &c.  are  the  normals  to  the  surface  at 
those  points,  and  as  each  is  intersected  by  its  con- 
secutive normal,  the  locus  TT'  .  .  .  of  these  in- 
tersections is  a  curve.  The  locus  too  of  the  nor- 
mals PT,  P'T',  &c.  themselves  form  a  surface, 
throughout  perpendicular  to  the  proposed;  this 
surface,  thus  generated  by  the  motion  of  a  straight 
line  PT  along  the  curve  PP'  .  .  .  and  each  position  intersecting  its 
consecutive  position,  is  obviously  a  developable  surface ;  one  of 
whose  edges  is  the  line  of  curvature  P  P'  .  .  .  and  the  other  the  line  of 
centres  T  T'  .  .  .  which  latter  is  called  the  edge  of  regression  of  the 
developable  surface.  Proceeding,  in  like  manner,  along  the  other 
line  of  curvature  through  P,  we  have  another  developable  normal 
surface,  whose  edge  of  regression  is  the  locus  of  the  centres  of  cur- 
vature belonging  to  this  second  line  of  curvature.  Applying  similar 
considerations  to  every  point  on  the  surface  (S),  we  shall  thus  have 
ao  infinite  number  of  developable  normal  surfaces  at  right  angles  to 


THE    DIFFEUENTIAIi    CALCUL08.  191 

each  other,  and  which  will  obviously  form  together  two  continuous 
volumes,  and  the  edges  of  regression  will,  in  like  manner  form  two 
continuous  surfaces,  or  sheets,  being  the  locus  of  all  the  centres  of 
curvature.  These  surfaces,  therefore,  bear  the  same  relation  to  the 
original  surface,  as  that  which  in  plane  curves  we  have  called  the 
evolute  bears  to  the  involute. 

It  would  be  quite  incompatible  ^vith  the  pretensions  of  this  little 
volume  to  extend  any  further  our  inquiries  into  the  properties  of  lines 
of  curvature.  For  more  detailed  information  respecting  these  re- 
markable lines,  the  student  must  study  the  illustrious  author  by  whom 
they  were  first  considered,  Monge,  in  his  Application  de  P Analyse  d 
la  Geometrie,  a  work  abounding  with  the  most  profound  and  beautiful 
speculations  on  the  subject  of  curve  surfaces  and  curves  of  double 
curvature,  and  which,  together  with  the  Developpements  de  Geometrie 
of  Dupin,  constitute  a  complete  body  of  information  on  a  very  at- 
tractive and  important  branch  of  mathematical  study,  the  cultivation 
of  which,  however,  has  been  almost  entirely  neglected  hitherto  in  this 
country.* 

Radii  of  Spherical  Curvature. 

(168.)  We  have  already  seen  that  the  radii  of  spherical  curvature, 
or  simply  the  radii  of  curvature  at  any  point  of  a  surface,  are  identi- 
cal to  the  radii  of  the  principal  sections  through  that  point,  and  have 
given  tolerably  commodious  formulas  for  the  calculation  of  these  radii 
when  the  axes  to  which  the  surface  is  referred  originate  at  the  proposed 
point,  the  plane  of  xij  being  coincident  with  the  tangent  plane,  and 
the  Eixis  of  z  with  the  normal  at  that  point.  We  have  also  seen  that 
when  these  radii  are  determined,  a  paraboloid  may  also  be  determined, 
having  its  vertex  at  the  proposed  point  and  its  curvature  in  all  direc- 
tions round  that  point  and  in  its  immediate  vicinity,  the  same  as  the 
curvature  of  the  surface ;  so  that  be  the  surface  ever  so  complicated, 
its  curvature  at  any  particular  point  will  be  correctly  presented  to  us 
by  the  vertex  of  a  determinable  paraboloid.  All  this,  however,  sup- 
poses the  radii  of  curvature  of  the  surface  at  this  point  to  be  known  • 

*  The  only  English  Mathematician,  I  believe,  who  has  produced  public  proof 
of  his  having  given  much  attention  to  these  inquiries,  is  Mr.  Davits  of  Bath, 
whose  papers  on  surfaces,  &c.  in  Leyhourri's  Repository,  I  have  already  had  occa- 
sion  to  refer  to. 


192  THE  DIFFERENTIAL  CALCULUS. 

it  remains,  therefore,  to  show  how  these  radii  may  be  determined, 
whatever  be  the  position  of  the  coordinate  axes. 

PROBLEM    IV. 

(169.)  Given  the  coordinates  of  a  point  on  a  curve  surface  to  de- 
termine the  radii  of  curvature  at  that  point. 

Let  (ar,  y,  z,)  be  the  point  on  the  surface,  and  (x',  ij\  z',)  either  of 
the  sought  points  on  the  normal  corresponding  to  the  centres  of  cur- 
vature, then  the  radius  R  from  either  will  be  given  by  the  expression 

R3    =    (,,'  _  ^)2   +    (y'  _  yY  +   (^'  _  zf. 

Since  at  the  proposed  point  the  equations  (1)  and  (3),  at  art,  (162), 
must  exist  simultaneously  with  this,  we  have,  by  substituting  in  this 
the  values  of  {x'  —  x),  (y  —  y)  as  given  by  (1), 

R  =  (2/  _  z)  ^/  i+p'^+  q'^. 

o/u 
Now,  if  from  (4),  (5)  we  eliminate  the  unknown  -p,  we  have 

(z  —  zj  (*•'  t'  —  «'")  +  (2  _  z')  { (1  +  q")  r'  —  2p'  q'  a'  + 
{l-\-p")t'\-\-{l+p"-\-q")=0, 
or,  putting  according  to  Monge 
g  =  r'  i'  —  s'3 
^  =  (1  +  q')  r'  —  2p'  q'a'  +  (1  +  /)'')  t' 

the  equation  for  determining  z  —  s'  becomes 

(z-zr  +  l{z-.z')+^  =  0 (1), 

and  the  roots  of  this  substituted  in  the  equation 

R  =  (2  _  2')  k, 

give 

R=  ^{h±  ^/"^m^  ....  (2) 


h  ±  ^h'  —  ^gk" 
(170.)  Thus  the  radii  of  curvature  are  determined,  and  the  direc- 
tions of  the  lines  of  curvature,  and  therefore  also  of  the  principal  sec- 


THE    DIFFERENTIAL    CALCULUS.  193 

tions  are  determined  by  Problem  III. ;  consequently,  the  radius  of  cur- 
vature of  an  oblique  section,  any  how  inclined  to  coordinate  planes, 
any  how  situated  with  respect  to  the  surface,  may  now  be  determined 
by  help  of  the  formulas  (9)  and  (1)  at  pages  181  and  184.  It  ap- 
pears from  (3)  that  the  surface  will  be  convex  or  concave  in  the  di- 
rection of  a  line  of  curvature  in  the  immediate  vicinity  of  the  point, 
according  as  g-  7  0  or  g-  Z  0.  If  g-  =  0  the  equation  (2)  shows  that 
one  of  the  radii  will  be  infinite. 

When  the  functions  of  x,  y,  z,  represented  by  p\  q',  r',  s',  t\  are 
complicated,  the  expressions  just  deduced  for  the  radii  of  curvature 
will  obviously  be  complicated  in  the  extreme.  They  are,  however, 
easily  manageable  when  the  proposed  surface  is  of  the  second  order, 
as  Dupin  has  shown  in  his  Developpements  for  both  classes  of  these 
surfaces.  We  shall  here  give  the  solution  for  surfaces  which  have 
not  a  centre,  that  is  for  paraboloids  ;  the  process  for  the  other  class, 
or  for  central  surfaces,  being  exactly  the  same  but  rather  longer. 

PROBLEM    V. 

(171.)  To  determine  the  radii  of  curvature  at  any  point  in  a  para- 
boloid. 
The  general  equation  of  paraboloids  being 

|=^  +  2.  =  0. 


we  have 


X 


'    —  y       .     1    _L  «.'2  _L    ^'2  _     ^  _1_    !f 


P'=-X'9'  =  -F-*-^+^"  +  ^"  =  P  +  B^+^=^ 


Hence,  generally,  whatever  be  the  paraboloid,  we  have,  for  the  co- 
efficients in  equation  (1)  above,  the  values 

^  =  _A  +  B-2.,|=AB(^  +  |  +  l)' 
25 


^94  THE    DIFFERENTIAL    CALCULUS, 

and  for  R  we  have 


The  sum  of  the  two  radii  are,  therefore, 


R  +  r  =  V^  +  |^+  1  X  (A  +  B-2Z) 

but  (124)  the  first  of  these  factors  is  the  reciprocal  of  the  cosine  of 
the  inclination  a  of  the  normal  at  the  point  {x,  y,  z,)  to  the  axis  of  z, 

.-.  (R  +  r)  cos.  a  =  A  +  B  —  2z, 
which  is  the  expression  for  the  sum  of  the  projections  of  the  radii  of 
curvature  on  the  axis  of  2;  A,  B  being  the  semi-parameters  of  the 
sections  on  the  planes  o^  xy,  yz.     If  the  point  be  at  the  vertex,  then 
X  =  0,  t/  =  0,  z  =  0,  and  the  values  of  R  then  become 


.'.  R  =  A,  r  =  B, 

and  these  are  also  the  radii  of  curvature  of  the  two  parabolic  sections 
on  the  planes  o^xy,  yz  (94),  so  that  these  sections  which  we  have  al- 
ready called  the  principal  sections  in  the  Analytical  Geometry,  are 
really  the  principal  sections,  or  those  of  greatest  and  least  curvature. 
A  similar  process  leads  to  similar  inferences  for  central  surfaces  of 
the  second  order. 


CHAFTERIV. 

ON  TWISTED  SURFACES.* 

(172).  We  have  already  stated  (148)  a  twisted  surface  to  be  one 
whose  generatrix  is  a  straight  line  moving  in  such  a  manner  along  its 
directrices  that  it  continually  changes  the  plane  of  its  motion. 

♦  This  is  the  class  of  surfaces  called  by  the  French  Surfaces  Gattches,  and 
which,  together  with  the  class  of  developable  surfaces,  they  include  under  the 


THE  DIFFERENTIAL  CALCULUS.  19i 

The  present  chapter  will  be  devoted  to  the  consideration  of  this 
class  of  surfaces.  Proceeding  from  the  simpler  kinds  to  the  more 
general,  we  shall  first  examine  the  surfaces  whose  directrices  are 
straight  Hnes  as  well  as  the  generatrices,  then  those  having  one  of  its 
directrices  a  curve,  afterwards  those  having  two  curvilinear  directrices, 
and  lastly  those  having  three  directrices  of  any  kind. 

Twisted  Surfaces  having  Rectilinear  Directrices  only. 

PROBLEM  I. 

(173.)  To  determine  the  surfaces  generated  by  a  straight  line  mo- 
ving parallel  to  a  fixed  plane,  and  along  two  rectilinear  directrices  not 
situated  in  one  plane. 

Let  the  fixed  plane,  called  the  directing  plane,  be  taken  for  that  of 
xy^  and  the  plane  parallel  to  the  two  directrices  for  that  of  xs ;  then 
the  equations  of  these  directrices  will  be 

,,.  x  =  az-\ra)         .   ix  =  a!z  +  a'  , 

(1) y  =  ^  p»d  {  y  =  ^'  •  •   •  •   (2;. 

and  the  generatrix  being  parallel  to  the  plane  of  xj/  will  be  represented 
by  the  equations 

s  =  6, 7/  =  mar  +  »  .  .  .  .   (3). 
As  this  fine  has  always  a  point  in  common  with  (1),  the  four  equations 
(1),  (3)  exist  together,  therefore,  eliminating  x,  ?/,  ^,  we  have,  among 
the  variable  parameters,  the  relation 

^  =  m  (a6  +  a)  +  n  .  .   .   .   (4), 
the  parameters  a,  a,  ^,  being  fixed  by  the  position  of  the  directrices, 
but  the  others  variable. 

In  like  manner,  since  the  line  (3)  has  also  always  a  point  in  com- 
mon with  (2),  the  four  equations  (2),  (3)  exist  together,  therefore, 
eliminating  x,  i/,  «,  we  get  for  a  second  relation  among  the  three  ar- 
bitrary parameters  the  equation 

general  name  of  Surfaces  RigUes,  expressive  of  their  mode  of  generation  by 
straight  line  generatrices.  There  has  just  appeared,  in  Leyhourii's  Repository, 
No.  22,  a  very  masterly  inquiry  into  the  history  of  these  surfaces,  from  the  pen  of 
Mr.  Davits,  wherein  the  claims  of  the  English  to  the  first  consideration  of  "rule 
surfaces  "  is  fully  established. 


196 


THE    DIFFERENTIAL    CALCULUS. 


13'  =m  {a'b  +  a')  +  n  .   .  .  .   (5). 
By  means  of  the  two  relations  (4)  and  (5)  among  the  parameters 
which  enter  (3),  we  may  eliminate  them  and  thus  obtain  the  sought 
equation  in  x,  y,  z.     Subtracting  each  from  (3),  we  have 

y  —  /3  =  m  (x  —  az  —  a) 
y  —  ^'  =  m  {x  —  a'z  —  a'), 
eliminating  m  we  obtain,  finally, 

(a  —a')  yz  +  {a  —  a')y-\-  (a'i3  —  a/3)  c  +  (/3'  —  (3)  x 
=  a/S'  —  a'/3  .   .   .   .    (6) 

for  the  equation  of  the  surface,  which  is  therefore  of  the  second  order. 
Let  us  now  inquire  what  particular  kind  of  surfaces  of  the  second  or- 
der this  equation  includes.  By  applying  the  criteria  (3)  (^^^naZ.  Geom.) 
we  find  that  the  surfaces  are  not  central,  they  must,  therefore,  be  pa- 
raboloids. By  putting  a;  =  A;  we  find  in  the  resulting  equation  for 
any  section  parallel  to  the  plane  of  yz,  that  the  squares  of  the  variables 
are  absent,  therefore,  {Anal  Geom.)  these  sections  are  all  hyperbolas. 
We  infer,  therefore,  that  the  surface  (6)  is  always  a  hxiperholic  para- 
boloid. If,  in  the  equation  (6)  we  make  z  equal  to  any  constemt 
quantity,  the  equation  will  always  be  that  of  a  straight  line,  being  in- 
deed necessarily  one  of  the  positions  of  the  generatrix ;  also,  if  we  put 
y  equal  to  any  constant  quantity,  we  find  that  every  section  parallel 
to  the  plane  of  xz  is  a  straight  line,  so  that  through  every  point  on  the 
surface  of  a  hyperbolic  paraboloid  there  may  be  drawn  two  straight 
lines,  their  assemblage  constituting  two  distinct  series  situated  in 
two  distinct  series  of  parallel  planes,  and  hence  there  are  two  distinct 
ways  in  which  the  surface  may  be  generated  by  the  motion  of  a  straight 
line,  but  not  more  than  two  ways,  since  the  equation  (6)  represents  a 
straight  line  only  on  the  two  hypotheses  assumed  above ;  and  as  no 
two  of  the  positions  of  the  same  generatrix,  however  close,  can  be  in 
the  same  plane,  the  hyperbolic  paraboloid  is  a  twisted  surface. 

(174.)  We  may  show  at  once  by  setting  out  with  the  equation  of 
the  hyperbolic  paraboloid,  that  two  straight  lines  pass  through  every 
point  on  its  surface,  and,  moreover,  that  these  lines  are  both  in  the 
tan  went  plane  at  that  point.  Thus  the  equation  of  the  surface  is  {Anal. 
Geom.) 

p^  —  pY  =  PP'z  ....  (1), 


THE  DIFFERENTIAL  CALCULUS.  197 

and  that  of  the  tangent  plane  through  (x',  y',  z',) 

2pxx'  —  ^p'yy'  ~  pp'  (2  +  2')  .  .  .  .  (2), 
the  relation  among  the  coordinates  x,  y',  2',  of  the  point  of  contact 
being  of  course 

px'^  —p'y""  =  pp'z'  ....   (3). 
Adding  together  equations  (1)  and  (3)  and  subtracting  (2)  from  the 
sum,  there  results 

which  is  the  condition  necessary  to  be  satisfied  for  every  projected 
point  (x,  y,)  common  to  the  surface  (1)  and  the  plane  (2),  seeing  that 
it  has  resulted  from  the  combination  of  their  equations.  Such  con- 
dition being  satisfied  by  every  point  in  the  Unes  represented  by  the 
equation 

y-y'=  ±  {x-x')V^, 

it  follows  that  the  lines  of  which  these  are  the  projections  are  common 
to  both  surface  and  tangent  plane,  so  that  the  tangent  plane  cuts  the 
surface  according  to  two  straight  lines  passing  through  the  point  of 
contact. 

PROBLEM    II. 

(175.)  To  determine  the  surface  generated  by  the  motion  of  a 
straight  line  along  three  others  fixed  in  position,  so  that  no  two  of 
them  are  in  the  same  plane. 

Let  us  first  consider  the  case  in  which  the  three  directrices  are  all 
pEirallel  to  the  same  plane. 

Assume  the  axes  of  x  and  y  in  this  plane  passing  through  one  of 
the  directrices  (B),  and  parallel  to  the  other  two  (B),  (B ").  Let  the 
axis  of  x  coincide  with  (B),  and  the  axis  ofy  be  parallel  to  (B'),  and 
let  the  axis  of  2  be  drawn  to  pass  through  both  (B)  and  (B"),  then 
the  equations  of  the  directrices  will  be 

(B)  y  =  0,     z  =  0 

(B')  x  =  0,    z  =  h 

(B")  y  =  ax,  z  =  k 

and  the  equation  of  the  generatrix  in  any  position  will  be 
X  =  mz  +  p,  y  =  nz  +  q  .  .  .  .  (1). 


198  THE  DIFFERENTIAL  CALCULUS. 

As  this  line  has  always  a  point  in  common  with  the  directrices,  all 
these  equations  exist  together.  Hence,  eUminating  x,  y,  z,  we  have, 
among  the  variable  parameters  m,  n,  p,  q,  the  relations 

q  =  0,  mh  ■]-  p  =  0,  Ilk  =  a  {mk  -\-  p)  .  .  .  .   (2). 
Eliminating  the  variable  parameters  from  (1)  and  (2),  we  have 
a  {k  —  h)  xz  =  hj  {z  —  h) 

for  the  equation  of  the  surface  sought,  and  which  we  find,  by  applying 
the  same  tests  as  in  last  problem,  to  be  the  same  surface,  viz.  the  hy- 
perbolic paraboloid. 

(176.)  Suppose,  now,  that  the  three  directrices  are  not  all  parallel 
to  the  same  plane,  then,  taking  any  point  in  space  for  the  origin,  and 
parallels  to  the  directrices  for  axes,  the  equations  of  these  will  be 

(B)  X  =  a,    ij  =  13 

(B')  z  =  y,    X  =  a 

(B")  y  =^l3\z  =  y 

and  the  equation  of  the  generatrix  will  be 

X  =^  mz  +  p,  y  =  nz  -^  q  .  .  .  .   (1), 
which,  since  it  has  a  point  in  common  with  (B),  gives  rise  to  the  con- 
dition 

m  n  ^ 

and  having,  at  the  same  time,  a  point  in  common  with  (B),  and  an- 
other in  common  with  (B"),  we  have  the  additional  conditions 

a  =  my  +  p,  f3'  =  ny  -^  q  .  .   .  .   (3). 
Eliminating  now  the  arbitrary  parameters,  m,  n,p,  q,  by  means  of  (1 ), 
and  these  equations  of  condition,  we  shall  arrive  at  the  equation  of  the 
surface.     The  equations  (3)  give,  in  conjunction  with  (1), 

X  —  a             M  —  B' 
m  = ,  n  = — 

z  —  y  z  —  y 

p  =  a  —y ,  g  —  \3  —  ^  -,, 

•^  2  —  y  z  —  y 

which  values,  substituted  in  (2),  give 

(r  —  r')  a^  +  (/3'  —  ^)  a:s  +  (a  —  a')i/z  ) 

+  a/3'y  —  a^y'  J 


THE  DIFFERENTIAL  CALCULUS.  199 

for  the  equation  of  the  surface.  By  applying  the  usual  criteria, 
{Anal.  Geom.)  we  find  that  the  surface  must  be  a  hyperboloid,  and 
as  the  squares  of  the  variables  are  all  absent  from  the  equation,  no 
intersection  {Anal.  Geom.)  can  possibly  be  an  imaginary  curve  ;  hence 
the  surface  must  be  a  hyperboloid  of  a  single  sheet,  and  it  is  obviously 
twisted,  since  the  generatrix  constantly  changes  the  plane  of  its 
motion. 

(177.)  We  may,  as  in  the  preceding  problem,  by  commencing 
with  the  equation  of  this  surface,  show  that  through  every  point  on  it 
two  straight  lines  may  be  drawn,  and  that  they  will  both  be  in  the 
tangent  plane  through  the  point.  Thus  the  equation  of  the  surface 
is 

^+£--4=1  —  (1)' 

a^       V        c^ 
and  that  of  the  tangent  plane  through  (a?',  ?/',  z) 

the  relation  among  x',  y',  s',  being  fixed  by  the  equation 

_/2  „/2  ^2 

a^        0^        r 
Adding  together  equations  (1)  and  (3),  and  subtracting  twice  equa- 
tion (2)  from  the  result,  we  have 

{:c-x'y        {y-yj         {z  -  z'f  _ 

— -. +  — i;i -. 1 W, 

a  relation  which  must  have  place  for  every  point  common  to  both  the 
surface  and  the  tangent  plane. 

Also,  subtracting  (3)  from  (2) 

x'{x-xf)       y'{y-y')        2'{z-z')  _ 

a^         "^  6^  c='  "  .  .  .  .  V  ;• 

Now,  in  order  to  ascertain  whether  the  points  fulfilling  these  condi- 
tions can  lie  in  a  straight  line,  let  us  combine  them  with  the  equations 
of  a  straight  line  through  {x',  y',  z',)  viz. 

X  _  x'  =  o'  (2  —  Z'),  y—y'  =  b'{z—z')    .    .    .    .    (6). 

Substituting  in  the  equations  (4)  and  (5)  these  expressions  for  x  —  x', 
y  —  y\  we  have, 


200  THE  DIFFERENTIAL  CALCULUS. 

^  ^     a-  W         c^ 

'  '    a^         h^         c^ 

aV       6Y  _^  2'  _ 

these  relations,  therefore,  must  exist  among  the  constants  in  (6),  for 
it  to  be  possible  for  that  line  to  belong  to  the  surface.  From  the 
second  of  these  we  readily  deduce  a  rational  value  of  a'  which,  sub- 
stituted in  the  first,  h'  will  be  given  by  the  solution  of  the  quadratic, 
which  will  furnish  two  values,  so  that  two  lines  passing  through  the 
point  of  contact  maybe  drawn,  that  shall  be  common  to  both  the 
surface  and  the  tangent  plane. 

Twisted  Surfaces  having  but  one  Curvilinear  Directrix. 

( 178.)  In  surfaces  of  this  kind  the  generatrix  moves  along  a  straight 
line  and  a  curve,  remaining  constantly  parallel  to  a  fixed  plane  called 
the  directing  plane.  Such  surfaces  are  called  conoids,  and  that  they 
are  twisted  surfaces  is  plain,  because  a  plane  to  pass  tlirough  two 
positions  of  the  generatrix  must  pass  through  the  rectilinear  directrix, 
and  become,  therefore,  fixed,  so  that  it  carmot  be  moved  round  one 
position  without  ceasing  to  pass  through  two.  The  directing  plane 
is  usually  taken  for  that  of  xy,  the  origin  being  at  the  point  where  the 
straight  directrix  pierces  it. 

PROBLEM   III. 

(179.)  To  determine  the  general  equation  of  canoidal  surfaces : 
Let  the  equations  of  the  straight  directrix  be 
X  =  mz,  y  =  nz  ,  .  .  .  (1), 
and  those  of  the  curvilinear  directrix, 

F  {x,  y,  z)  =  0,f{x,  J/,  2)  -  0  ...  .  (2), 
The  equation  of  the  generatrix,  being  in  every  position  parallel  to  the 
plane  of  ary,  must  always  be  of  the  form 


THE  DIFFERENTIAL  CALCULI'S. 


201 


z  =  a,  y  =^  (3x  +  y  .  .  .  .  (3), 

ft  and  /3  being  variable  parameters. 

As  this  line  has  always  a  point  in  common  with  the  line  (1),  their 
equations  exist  together ;  hence,  eliminating  x,  y,  z,  by  means  oi 
these  four  equations,  we  have  the  condition 

no,  =  f3ma  +  y  or  y  =  na  —  (3ma, 
so  that  the  equations  (3)  of  the  generatrix  become 

z  =  a,y  —  na  =  /3  (x  —  ma)  ....   (4) ; 
but  this  same  line  has  also  a  point  in  common  with  the  curve  (2) ; 
hence,  eliminating  x,  y,  z,  by  means  of  the  four  equations  (2),  (4), 
we  have  an  equation  containing  only  constants  and  the  variable  para- 
meters a,  /3,  which  equation,  solved  for  a,  gives 

a  =  9  :  j3  .  ,  .  .   (5).* 
But,  by  equations  (4), 

^       y  —  nz 

a  =  z,^  =  2 ; 

X  —  mz 

faence,  by  substitution  in  (5), 

y  —  nz^ 

s  =  <P  {- , 

X  —  mz 

which  expresses  the  general  relation  among  the  coordinates  of  any 
point  of  the  generatrix  in  any  position,  therefore  this  is  the  general 
equation  of  a  conoidal  surface. 

(180.)  If  the  straight  directrix  coincide  with  the  axis  of  2,  then 
m  =  0,  w  =  0,  and  the  conoid  is  represented  by  the  general  equation 

z  =  (p  (|), 

whether  the  axis  of  2,  or  the  straight  directrix,  be  perpendicular  to  the 
directing  plane  or  not ;  if  it  is  perpendicular,  the  conoid  is  called  a 
right  conoid.  In  these  cases  the  equations  of  the  generatrix  are 
simply 

z  =  a,y  =  I3x. 
(181.)  As  an  example,  let  it  be  required  to  find  the  equation  of  the 
inferior  surface  of  a  winding  staircase,  the  aperture  or  column  round 
which  it  winds  being  cylindrical. 

*  0  :  /J  is  the  same  as  (p0  or  a  function  of  0.  Ed. 

26 


THE  DIFFERENTIAL  CALCITLUS. 

To  conceive  the  generation  of  this  surface,  let  us  suppose  a  rect- 
angle to  be  rolled  round  a  vertical  column,  which  it  just  embraces, 
the  line  which  was  the  diagonal  of  the  rectangle  will  then  become  a 
winding  curve  called  a  helix,  and  it  will  make  just  one  turn  round  the 
column,  its  horizontal  projection  being  a  circle ;  if  immediately  above 
this  another  equal  rectangle  be  applied  to  the  column,  the  vertical 
edges  when  brought  together  being  in  a  line  with  those  of  the  first, 
the  diagonal  of  this  will  form  a  continuation  of  the  helix,  and  in  this 
way  will  be  exhibited  the  trace  of  the  edge  of  the  surface  in  question 
on  the  vertical  column,  or  the  curvilinear  directrix  ;  the  other  direc- 
trix is  the  axis  of  the  cylinder,  the  directing  plane  being  horizontal. 

Now  for  every  point  in  the  diagonal  of  a  rectangle  the  abscissa  has 
a  constant  ratio  to  the  ordinate,  the  axes  being  the  sides  including  the 
diagonal,  so  that,  reckoning  from  the  foot  of  the  helix,  the  circular 
abscissas  and  vertical  ordinates  corresponding  are  in  a  constant  ratio. 
Hence,  taking  the  centre  of  the  cylindrical  base  for  the  origin  and 
drawing  the  axis  of  y  through  the  foot  of  the  helix,  calling  h  the 
height  and  2'jn'  the  base  of  one  of  the  rectangles,  or  of  the  cylinder, 
we  shall  have  for  each  point  of  the  helix,  these  relations,  viz. 

_3   I      "         o  .       s     z    _    h 

ar  +  y  =  r^,  X  ~  r  sm.  — ,  —  =  -; —  ....  (1), 
^  r     s        2<irr 

and  for  the  generating  line  the  equations 

z  =  a,y  =  I3x  .  .   .  .  (2). 

If  from  the  two  last  of  (1)  we  eliminate  the  arc  s  we  shall  have  the 

following  equations  of  the  projections  of  the  curve 

2irz 
X  -\-  y"  =  r^,  X  =  r  sin.  (-r-)  ....  (3), 

eliminating  x,  y,  z,  from  the  equations  (2),  (3)  we  have 

sm.  (2*  j-), 


n/1  +  iS^  '      h' 

in  which  equation  if  we  substitute  for  a  and  /3  the  values  z  and  — 

X 

given  by  (2),  we  shall  obtain,  finally. 


sin.  (2*  -J-)  or  -  =  tan.  (2ir  -) 


which  is  the  equation  of  the  surface,  that  is  of  the  twisted  helixoid. 


THE    DIFFERENTIAL    CALCULUS.  203 

( 182. )  It  remains  to  determine  the  differential  equation  of  conoidal 
surfaces.  In  order  to  this  we  must  eliminate  the  arbitrary  function 
tp  in  the  equation 

y  —  nz 

^  =  <p  r — —h 

X  —  ntz 
by  differentiating,  as  in  the  several  similar  cases  in  Chapter  II.,  we 
thus  obtain  the  equation 

p'  _  p'  {my  —  nx)  —  {y  —  nz) 

q'        q'  [my  —  nx)  +  (x  —  mz) 
which  reduces  to 

p'  {x  —  mz)  +  9'  (y  —  nz)  —  0, 

or  when  the  conoid  is  right  simply  to 

p'x  +  q'y  =  0, 

because  then  m  =  0,  »  —  0. 

(183.)  The  same  results  may  be  at  once  obtained  from  the  con- 
sideration of  the  tangent  plane  ;  for  {x',  y',  s',)  being  any  point  on  the 
surface,  the  equation  of  the  tangent  plane  is 

z  —  z'=p{x  —  x')  +  q'  {y  —  y'), 
which  touches  the  surface  along  the  generatrix  through  {x,  y\  z'), 
and  this  being  every  where  at  the  same  distance  z'  from  the  horizon- 
tal plane,  it  follows  that  if  in  the  above  equation  we  put  z  =  z'  the  re- 
sult 

p'  {x  —  x)  +  q'  (y  —  y')  =  0 
will  express  the  relation  between  the  x,  y  of  every  point  in  this  gene- 
ratrix.    But  at  that  point  where  it  cuts  the  straight  directrix,  the  x,  y 
have  the  relation 

X  =  mz',  y  =  nz'f 
so  that,  by  substitution,  we  have 

p'  {mz'  —  x')  +  q'  {nz'  —  y')  =  0, 

for  the  relation  among  the  coordinates  of  every  point  (x',  y',  z',)  on  the 
surface,  which  agrees  with  that  deduced  above. 

Twisted  Surfaces  having  Curvili^iear  Directrices  only, 
(184.)  We  now  proceed  to  consider  those  surfaces  which  cannot 


204  THE  DIFFERENTIAL  CALCULUS. 

have  a  rectilinear  directrix,  or  rather  those  whose  directrices  may  be 
any  lines  whatever.     We  shall  first  suppose  two  directrices. 

PROBLEM  IV. 

(185.)  To  determine  the  general  equation  of  surfaces  generated 
by  a  straight  line  which  moves  along  any  two  directrices  (D),  (D') 
whatever,  and  continues  at  the  same  time  parallel  to  a  fixed  plane. 

Taking  as  before  the  directing  plane  for  that  o£  xy,  the  equation  of 
the  generatrix  in  any  position  will  be 

z  =  a,  y  =  I3x  -\-  y  .  .  .  .   (1), 

the  parameters  all  varying  with  the  varying  positions  of  the  generatrix. 

Let  now  the  equations  of  the  two  fixed  directrices  be 

(D)  F  {x,  y,  z)  =  0,f{x,  y,z)  =  0  .  .  .  ,  (2) 

(D')         F,  {x,y,  z)  =  0,/  {x,y,z)=0  .  .  .  .   (3). 

Then  the  condition  is,  first  that  the  generatrix  meets  (D),  or  that 
their  equations  (1),  (2)  exist  together;  hence  by  eliminating  the  co- 
ordinates of  the  common  point  from  these  four  equations,  we  shall 
obviously  obtain  an  equation  containing  only  constants  and  the  varia- 
ble parameters  a,  (3,  y,  that  is  to  say,  we  shall  obtain  among  these 
parameters  a  relation 

*  («,  ^,  rO  =  0. 

Proceeding  in  the  same  manner  with  the  equations  (1),  (3)  which 
also  exist  together  for  a  certain  point,  we  obtain  a  second  relation 

Y  (a,  ^,  7,)  =  0. 

By  means  of  these  two  equations  we  may  eliminate  any  one  of  the 
parameters  ;  therefore,  eliminating  first  y  and  then  ^,  we  have 

B  =  cpicc,  y  =  -^'.a ; 

hence,  substituting  for  these  variable  parameters  their  values  in  func- 
tions of  the  variable  coordinates  as  furnished  by  equation  (1),  we 
have,  for  the  general  relation  among  these  coordinates,  the  equation 

xy  =■  9:2  +  -^'.z  ....  (4). 

This  then  is  the  general  equation  of  all  surfaces  generated  as  an- 
nounced, whatever  be  the  form  of  the  directrices ;  when  these  forms 
are  given,  the  forms  of  9  and  -^  become  determinable  by  the  above 


THE  DIFFERENTIAL  CALCULUS.  206 

process,  and  then  the  general  equation  (4)  takes  the  particular  form 
belonging  to  the  individual  surface. 

(186.)  Let  us  now  determine  the  general  equation  of  these  surfa- 
ces in  terms  of  the  partial  differential  coefficients.  Putting  the  equa- 
tion (4)  in  the  form 

y  —  x(p:z  =  4.:r, 

the  ratio  of  the  partial  coefficients  of  each  side,  taken  relatively  to  x 
and  y,  will,  by  the  principle  in  (58),  be 

—  o:z        p      ,      •      v' 

— ~—  =  ^,  that  IS,  ^  =  —  (p:2, 

1  9  9 

an  equation  from  which  the  arbitrary  function  4^:5  is  eliminated.  Ap- 
plying the  same  principle  to  this  last  equation  we  have 

q^  ^     9     _  Pi 


dx  dy  q' 

that  is,  putting  according  to  the  usual  notation 

dx  ^  dx         dy  ^  dy 

q'r'  —  p's'    .    g's  —  p't'  _  p 

?^"^  ?  ?' 

whence 

g'V  —  2p'q's'  +  p'H'  =  0, 

an  equation  from  which  both  the  arbitrary  functions  are  eliminated, 
and  which  must  be  fulfilled  for  every  point  in  every  surface  generated 
as  in  the  problem,  whatever  be  the  directrices.  We  see  that  as  two 
arbitrary  functions  were  to  be  eliminated,  the  process  led  to  a  partial 
differential  equation  of  the  second  order. 

PROBLEM  V. 

(187.)  To  determine  the  general  equation  of  surfaces  generated 
by  the  motion  of  a  straight  line  along  three  curvilinear  directrices  (D) 
(DO,  (D"). 

We  shall  first  remark  that  the  motion  of  the  generatrix  is  entirely 
governed  by  these  conditions,  for  if  we  take  any  point  on  the  first  di- 
rectrix (D)  and  conceive  two  cones  whose  bases  are  (D'),  (D")  to 


206  THE  DIFFERENTIAL  CALCULUS. 

have  this  point  for  their  common  vertex,  these  cones  will  obviously 
intersect  each  other  in  all  the  straight  lines  that  can  be  drawn  from 
the  point  to  the  curves  (DO,  (D"),  the  positions  of  these  lines  are 
therefore  fixed  by  these  intersecting  cones,  and  these  are  fixed  by  their 
bases  ;  hence,  all  the  lines  that  can  be  drawn  from  the  point  to  the 
lines  (DO,  (D'O  are  determinate  both  in  number  and  position,  this 
being  true  for  every  point  in  (D),  it  follows  that  the  surface  generated 
by  all  these  lines  is  determinate,  and  it  is  now  required  to  find  its 
equation. 

As  there  is  here  no  directing  plane  the  equations  of  the  generatrix 
in  any  position  will  take  the  form 

X  =  az  +  f,  y  =  I3z  -^^  S  .  .  .  .  (1), 
and,  since  it  always  has  a  point  in  common  with  (D),  we  may  elimi- 
nate by  means  of  the  equation  of  (D)  combined  with  these,  the  coor- 
dinates of  that  point :  the  result  will  furnish  a  condition  among  the 
variable  parameters.  In  like  manner,  employing  the  equation  of 
(D')  we  shall  arrive  at  another  equation  of  condition,  and,  lastly,  the 
equation  of  (D'O  will  furnish  a  third  equation.  By  means  of  these 
three  equations  any  two  of  the  parameters  a,  j3,  y,  S,  may  be  ehrai- 
nated,  and  we  shall  obtain  three  equations  of  the  form 

13  =  (p:a,  y  =  -^la,  6  =  if:a. 
substituting  these  expressions  for  /3,  y,  S,  in  the  equations  (1)  we  have 

X  =  az  -{-  -^-.a,  y  =  z(p:a,  +  *:«, 

two  equations  which  have  place  for  every  surface  generated  as  pro- 
posed, the  functions  which  fix  the  directrices  being  quite  arbitrary.  If 
these  functions  are  known,  or  the  directrices  fixed,  we  may  then  eli- 
minate the  parameter  a  by  means  of  these  equations,  and  thus  deduce 
the  equation  of  the  individual  surface,  but  the  general  relations  among 
the  coordinates  for  all  the  surfaces  of  this  family  can  be  exhibited  only 
by  means  of  two  equations  as  above.  The  general  relation  among 
the  partial  differential  coefficients  belonging  to  all  this  family  of  sur- 
faces may,  however,  be  ascertained  in  a  single  equation  by  eliminat- 
ing, as  in  last  problem,  all  the  arbitrary  functions  by  successive  dif- 
ferentiation ;  this  will  lead  to  a  partial  differential  equation  of  the  third 
order,  for  which  see  JVLonge's  Application  de  V Analyse  a  la  Geometne^ 
p.  196. 


THE  DIFFERENTIAL  CALCULUS. 


207 


(188.)  We  shall  terminate  the  present  chapter  with  the  following 
example : 

On  the  opposite  sides  of  the  hori- 
zontal parallelogram  AB  DC  are  de- 
scribed two  vertical  semicircles,  and 
perpendicular  to  their  planes  is  drawn 
the  straight  line  OY  through  the  centre 
of  the  parallelogram  ;  taking  this 
straight  line  and  the  two  semicircles 
as  directrices,  it  is  required  to  find 
the  equation  of  the  surface  generated 
by  a  straight  line  moving  along  them. 
Let  the  axes  of  coordinates  be  the 
perpendicular  horizontal  lines  OX, 
OY,  and  the  vertical  OZ,  then  the  equations  of  the  three  directrices 
will  be 

2  =  0,3  =  0  ....  (1) 
7/  =  —  6,  (ar  —  o)2  +  2?  =  r"  .  .  .  .  (2) 
y  =  +  b,{x  +  ay  +  z'^r'  .  •   .   .   (3). 
The  equations  of  the  generatrix,  since  it  always  passes  through  a 
point  (/8,  0,  0)  in  the  axis  ofy,  will  take  the  forms 

x  =  a{y  —  (3),z  =  Y{y  —  (3)  .  .  .  .  (4), 
and  the  condition  to  be  fulfilled  by  this  line  is,  that  it  rests  on  each  of 
the  semicircles  ;  or  that  at  certain  points,  x,  y,  z,  are  the  same  in  the 
equations  (2),  (4)  and  (3),  (4)  ;  hence,  eliminating  these  first  from 
(2),  (4),  and  then  from  (3),  (4),  we  have  these  relations  among  the 
variable  parameters,  viz. 

\a{b  +  ^)  +  a\''  +  y'{b  +  (3)' =  r' 
|a  (6  +  /3)  +  ap  -j-  f  (6  —  /3)2  =  r^ 
which,  by  subtraction,  give 

13  {ba?  +  aa  -\-  bjS^)  =  0. 

This  condition  is  satisfied  by  the  value  /3  =  0,  but  this  is  not  admis- 
sible, since  it  would  restrict  the  generatrix  to  pass  always  through 


(5) 

(6), 


the  origin,  and  have  no  motion  along  OY ;  hence,  dividing  by 


(3 

T' 


208  THE  DIFFERENTIAL  CALCULUS. 

we  have  the  relation 

«^  +  7^  +  y  =  0 (7), 

between  the  parameters  a,  y. 

Substituting  the  value  of  y^  given  by  this  equation  in  (5),  it  be-- 
comes 

(62  _  132^  aa  =  6  (r=  —  a^)   .  .  .  .  (8), 
and  by  means  of  these  equations,  together  with  those  of  the  genera- 
trix, we  may  readily  eliminate  the  parameters ;  thus  the  values  of  a 
and  y,  given  by  (4),  are 

and  these,  substituted  in  (7),  give 


p  =  y  -\- •••  a  = 


ax         '  '  b{xr^  -\-  z~y 

and  finally,  these  substituted  in  (8)  give  for  the  surface  the  equationi 

b{r^  +  ^)     _  ^3  =  ^3  (^  _  ^.)  (^^f!), 
'^  ox        '  aV 

which  is  the  same  as 

\axy-\-b{a^-\-  z')l^=  b^  r"  x"  ■\- b^r"  —  a")^. 


OHAFTfiR   V. 

ON  DEVELOPABLE  SURFACES  AND  ENVELOPES. 

(189.)  When  in  an  equation  between  three  variables 
F  (x,  y,  z,  a)  =  0, 
there  enters  an  arbitrary  constant  a,  that  equation,  by  giving  different 
values  to  a,  will  represent  so  many  different  surfaces  all  belonging  to 
the  same  family.  If  we  fix  one  of  these  by  any  determinate  value  of 
a,  another,  intersecting  this,  will  be  represented  by  changing  a  into 
a  +  h,  h  being  some  finite  value.  If  h  be  now  continually  dimi- 
nished, the  intersection  will  continually  vary,  and  will  become  fixed 


THE  DIFFERENTIAL  CALCULUS.  209 

only  when  the  varying  surface  becomes  coincident  with  the  fixed 
surface.  In  this  position  the  intersection  is  said  to  belong  to  con- 
secutive surfaces,  and  it  may  be  determined  both  in  form  and  position 
by  a  process  similar  to  that  employed  at  (105).  Thus  a  being  the 
only  variable  concerned  in  the  intersections,  let  tt  =  F  (r,  y,  2,  a), 
now  if  a  increase  by  h,  «'  =  F  (x,  y,  z,a  +  h)  which  developed  by 
Taylor's  theorem,  gives 


u 

= 

,     du 
da 

h-{- 

d?u 
da? 

4+- 

but, 

since  u  = 

0, 

,  therefore 

du 

doL    ^ 

(Pu 
da? 

.|  +  &c.  =  0; 

hence,  when  the  surfaces  are  consecutive,  that  is,  when  fe  =  0,  we 
have  the  following  equations  for  determining  the  curve  of  intersection, 
viz. 

du  =  Q\ 

du  =0\  .  .  .  .  (1), 

^  ) 

these,  therefore,  are  the  equations  of  the  curve,  which  is  the  intersec- 
tion of  the  surface  (1)  with  its  consecutive  surface. 

If  from  these  two  equations  we  eliminate  a,  the  result  will  be  the 
general  relation  among  the  coordinates  of  every  point  in  every  such 
consecutive  intersection  throughout  the  whole  family  of  surfaces,  this 
resulting  equation  will  therefore  represent  the  surface  which  is  the 
locus  of  all  these  consecutive  intersections.  This  locus,  moreover, 
touches  each  of  the  variable  surfaces  throughout  their  intersections ; 
for  differentiating  the  equation  F  =  0  of  any  one  of  the  variable  sur- 
faces, a  being  constant,  we  have 

du     ,    du     ,       ^    du    .    du     , 

and,  differentiating  the  equation  F  =  0  of  the  locus,  a  being  variable, 
we  have 

du         du^    ,   ,    du      da  _       du         du  du      da  _ 

dx   '^  dz  P  ^   da  '  dx~  ^'  lif^  'db  "^  '^  1^  "d^  ~  ^' 
which  equations  are  identical  to  those  above,  since 


du 


27 


210 


THE  DIFFERENTIAL  CALCULUS. 


and  therefore  each  pair  give  the  same  values  for  p'  and  q  consequently, 
at  the  points  common  to  both  surfaces  they  have  common  tangent 
planes.  Hence  the  locus  of  the  consecutive  intersections  touches, 
and  envelopes  all  the  variable  surfaces ;  it  is,  therefore,  called  by 
Monge  the  Envelope  of  these  surfaces. 

(190.)  If  the  envelope  be  formed  by  the  consecutive  intersections 
o( planes,  then,  since  from  what  has  been  just  proved,  the  envelope  is 
touched,  throughout  each  of  the  intersections  by  the  corresponding 
plane ;  this  envelope  is  such  that  the  tangent  plane  at  any  point 
touches  it  throughout,  the  rectilinear  generatrix  passing  through  that 
point ;  and  this  is  the  characteristic  property  of  a  developable  surface  . 
hence  a  developable  surface  may  be  considered  as  the  envelope  of  a 
family  of  planes  represented  by  the  general  equation 

z  =  Ax  +  By-l-B  .  .  .  .  (1), 
in  which  there  enters  a  variable  parameter  a. 

Now,  to  introduce  this  variable  parameter  in  the  most  general 
manner  possible  into  the  equation  (1),  we  ought  to  consider  each  of 
the  coefficients  A,  B,  C,  to  be  functions  of  it,  so  that  the  general  form 
will  be 

s  =/a  +  x(})a  +  yipa,* 

and  therefore  the  line  of  contact  (189)  or  generatrix  of  the  surface 
will  be  represented  by  the  equations 

0  =f'a-^xcfia  -\-  y-Jf^'a  )   '  '  '  '  \  J' 
When  the  forms  of  the  functions/,  cp,  4^,  are  fixed,  the  variable  para- 
meter a  may  be  eliminated,  and  the  resulting  equation  in  x,  y,  z,  will 
be  that  of  the  individual  surface  to  which  these  particular  forms  be- 
long.   The  equations  (2),  therefore,  may  be  considered  as  represent- 

*  Monge  says  the  general  equation  may  always  be  put  under  the  form 
z  =  a;0a  -\--  yil/a  -\-  a, 
which,  however,  seems  to  be  incorrect,  since  it  excludes  those  of  the  family  com- 
prehended in  the  equationf 

2  =  X(pa  -f-  y4"'^  +  c» 
and  which  evidently  generate  conical  surfaces,  whose  vertices  are  all  on  the  axis 
of  z,  at  the  distance  c,  from  the  origin.    The  form  in  the  text  includes  this  class 
of  equations,  for /a  may  be  constant  without  causing  <pa  or  xpa  to  become  so. 

t  Monge  in  this  observation  is  not  incorrect,  since  a  is  indeterminate,  one  of  it» 
values  will  necesBarily  be  equal  to  c.  Ed. 


THE  mPPERENTIAL  CAXCULU8. 


211 


ing  the  whole  family  of  these  surfaces,  a.  in  the  first  being  a  function 
of  X  and  y,  impUed  in  the  second. 

(191.)  1.  As  an  example,  suppose  it  were  required  to  determine 
the  developable  surface  generated  by  the  intersection  of  normal  planes 
at  every  point  in  a  curve  of  double  curvature. 

Representing  the  proposed  curve  by  the  equations 
y  =  Fx',z'  =fx'  ....  (1), 
the  general  equation  of  the  normal  plane  will  be  (129) 

,_..+|;(j_,')+i;i(._.-)=o (2), 

in  which  the  only  variable  parameter  is  x  ;  y'  and  z  being  determinate 
functions  of  it  given  by  the  equations  of  the  curve.  Hence,  difier- 
entiating  with  respect  to  x,  we  have 

-<'+l^  +  ^'  +  ^(^-^')+£^(-^'  =  «-'^'- 

Now  the  functions  of  x',  which  enter  the  equations  (2),  (3),  being 
given  by  (1),  we  may  eliminate  this  parameter  from  them,  and  the 
resulting  equation  in  x,  y,  z,  will  be  that  of  the  developable  surface 
required. 

(192.)  2.  As  a  second  example,  let  it  be  required  to  determine 
the  developable  surface  which  touches  and  embraces  two  given  curve 
surfaces. 

If  we  suppose  one  of  these  surfaces  to  be  a  luminous  surface  en- 
lightening the  other,  the  surface  which  we  .seek  will  obviously  em- 
brace all  the  rays  which  proceed  from  the  bright  surface  to  the  dark 
one,  and  the  curve  of  contact  on  this  latter,  will  separate  the  illuminated 
and  dark  parts. 

Let  the  equations  of  the  two  given  surfaces  be 

Fi  (x-„  7/.,  s,)  =  0,  F,  {x,,  y,,  2,)  =  0  ...  (1), 
then  the  equations  to  tangent  planes  to  each  will  have  the  form 

z  —  z,,=  p,  {x  —  x,)  +  qi{y  —  y.)  ...  (2) 
and 

2  _  2^  =  Pj,  (^  —  -^-i)  +  q-z  {y  —  y-i), 

and  for  these  planes  to  belong  to  both  surfaces,  their  equations  must 
be  identical,  that  is,  we  must  have  the  conditions 

Pi  ~p!»qi  =qi  '  •  -  (3), 


212  THE  DIFFERENTIAL  CALCULUS. 

«i  —  Pi^i  —  qiy,  =  22  ~  p^x,  —  q,y.2  .  .  .  (4). 
By  means  of  the  six  equations  marked,  five  of  the  coordinates  may 
be  determined  in  terms  of  the  sixth,  x  ;  hence,  if  these  functions  of  a-, 
be  now  substituted  for  their  values  in  the  remaining  equation,  we  shall 
obtain  a  result  containing  only  the  variable  parameter  Xi,  and  which 
will  consequently  represent  the  family  of  planes  which  generate  the 
developable  surface  sought.  Calling  this  result  P  =  0,  the  genera- 
trix of  the  surface  will  be  given  by  the  equations 

P  =  0,f  =0, 

axy 

from  which,  eliminating  r,,  we  have  the  equation  of  the  surface 
sought ;  and  this  equation,  combined  with  that  of  each  surface  sepa- 
rately, will  give  the  two  curves  of  contact. 

PROBLEM. 

(193.)  To  determine  the  differential  equation  of  developable  sur- 
faces in  general. 

The  general  equation  of  the  generating  plane,  arranged  according 
to  the  variable  coordinates  x,  y,  z,  of  any  point  in  it,  is 

z  =  p'x  4"  q'y  +  2  — p'x  —  q'y'  ...  (1) 
and  this  plane  remains  the  same  for  every  point  in  the  generatrix,  as 
well  as  for  the  point  {x',  y',  z'),  so  that  the  quantities 

p\q',z'—p'x'  —  q'y'  .  .  .  (2), 
remain  constant,  although  x',  y',  z',  all  vary,  provided  this  variation  is 
confined  to  the  rectilinear  generatrix,  for  which  y  is  always  a  function 
of  X,  but  not  else  ;  hence,  the  conditions  which  restrict  the  point  x', 
y',  z',  to  the  generatrix  on  which  it  is  first  assumed,  is,  that  the  dif- 
ferential coefficients  derived  from  (2),  y  being  considered  as  a  func- 
tion of  x,  are  all  0,  and  it  is  plain  that  if  any  two  be  0,  the  third  will 
be  0  also ;  hence,  differentiating  the  two  first,  we  have 

ax  ax 

where -j^  fixes  the  position  of  the  rectilinear  directrix  for  which  the 

cly 
expressions  (2),  remain  constant.     Eliminating,  then  — ,  we  obtain 

the  following  equation,  which  must  hold  for  every  directrix,  viz. 


THB  DIFFERENTIAL  CALCULUS.  213 

r't'  _  s'2  =  0  .  .  .  c3). 

This,  therefore,  is  the  equation  which  the  differential  of  the  equa- 
tion of  every  developable  surface  must  accord  with ;  or,  in  usual 
terms,  it  is  the  differential  equation  of  developable  surfaces  in  general. 

(194.)  We  shall  exhibit  another  method  of  obtaining  the  equation 
(3),  from  the  general  equations  (2),  art.  (190).  Differentiating  the 
first  of  these,  in  which  a  is  a  function  of  x  and  y  impUed  in  the  se- 
cond, we  have  the  two  partial  differential  equations 

da  ,    da,  . ,    da 

,  da  ,     da  da 

but,  in  virtue  of  the  second  of  the  equations  (2),  these  become 

p'  =  (pa,  q'  =  -^a, 

consequently,  p'  must  be  a  function  of  q',  and  may  therefore  be  re- 
presented by 

p'  =  ifq'. 

Eliminating  now  the  arbitrary  function  •»'  by  differentiation,  as  in 
(58),  we  have 

s  I 

as  before. 

(196.)  We  have  as  yet  considered  only  the  simplest  class  of  sur- 
faces, whose  intersections,  with  their  consecutive  surfaces,  are  given 
by  the  general  equations  (1),  art.  (189),  viz.  plane  surfaces,  the  in- 
tersections being  straight  lines.  It  is  obvious,  however,  that  what- 
ever be  the  surfaces,  the  intersections  are  still  given  by  the  equations 
(1),  and  the  envelope  of  these  surfaces,  found  by  eliminating  from 
them  the  arbitrary  parameter  a.  This  parameter,  however,  may 
enter  the  equation  of  any  particular  family  of  surfaces  in  an  infinite 
variety  of  different  forms  and  ways  ;  it  may  enter  into  only  one  of  its 
terms,  or  be  combined  with  several ;  a  simple  power  only  of  it  may 
enter,  or  a  compUcated  function,  and  still,  entering  only  as  a  parame- 
ter, the  general  equation,  under  all  these  changes,  will  still  preserve 
the  same  character,  and  represent  but  one  family  of  surfaces.  With 
the  envelope,  however,  it  will  be  different ;  this  depends  as  well  on 


214  THE  DIFFERENTIAL  CALCULUS. 

the  arbitrary  parameter,  as  on  the  variables  which  enter  the  general 
equation,  since  the  value  of  this  parameter  must  be  found  from  one 
of  the  equations  (1),  in  terms  of  a,  y,  2,  and  this  value  substituted  in 
the  other  for  the  equation  of  the  envelope.  Nevertheless,  since,  as 
just  observed,  the  individual  surfaces  represented  by  the  two  equa- 
tions (1),  for  every  particular  value  of  the  parameter  in  whatever  form 
it  may  enter,  is  always  of  the  same  degree,  it  follows  that  each  indi- 
vidual intersection,  (1),  will  uniformly  be  a  curve  of  the  same  order, 
and  which  will  change  its  order  only  when  the  order  of  the  surface 
changes.  This  curve  of  intersection  or  of  contact,  common  to  all 
the  envelopes  of  the  same  family  of  surfaces,  is  called,  hy  Monge^ 
the  characteristic. 

Considering  any  of  the  characteristics  (1)  separately,  we  may  in- 
quire what  are  the  points  in  which  it  is  intersected  by  the  consecutive 
characteristic  ;  and  the  method  of  determining  these  intersections  is 
analogous  to  that  already  explained  in  (105)  and  (189),  that  is,  we 
must  combine  with  the  equations  (1)  of  this  curve  their  differentials 
taken  relatively  to  a ;  hence,  the  consecutive  intersections  for  any 
particular  position  of  the  characteristic,  will  be  determined  by  the 
equations 

M  =  0 

du  

da 

d^u 

each  of  these  separately  represent  a  surface,  any  two  together  a  line 
common  to  both,  and  all  three  the  point  or  points  common  to  their 
intersection,  a  being  considered  constant.  By  solving  these  three 
equations  for  x,  ?/,  and  2,  we  shall  obviously  obtain  known  values  for 
the  coordinates  of  the  points  of  intersection  required,  which  of  course 
are  all  situated  on  the  envelope. 

Now,  if  from  the  three  equations  above,  we  eliminate  a,  we  shall 
have  two  equations  in  x,  y,  2,  existing  together,  which,  being  the  same 
for  the  intersections  of  every  pair  of  consecutive  characteristics  must 
represent  the  locus  of  these  intersections,  and  be  situated  on  the  en- 
velope. It  will  therefore  be  a  line  which  touches  and  encompasses 
all  the  characteristics,  in  the  same  manner  as  the  envelope  touches 


THE  DlFF'iSUENTIAL  CAXCULUS.  21^ 

and  embraces  all  the  enveloped  surfaces.  It  must  then  form  an  edge 
of  the  envelope,  or  the  line  ia  whioli  its  sheets  terminate,  and  it  is 
therefore  called,  by  jyionge,  the  edge  of  regression  of  the  envelope. 
In  the  developable  surfaces,  we  have  seen  that  the  characteristic  is  a 
straight  line,  and  the  consecutive  intersections  of  the  characteristic, 
in  every  position,  obviously  form  the  edge  which  limits  the  locus  of 
the  characteristics,  that  is,  the  developable  surface. 

The  consideration  of  envelopes,  characteristics,  and  edges  of  re- 
gression, have  been  successfully  employed  by  JMonge,  and  succeed- 
ing writers,  to  remove  several  difficulties  in  the  higher  departments 
of  the  integral  calculus,  that  do  not  appear  to  be  otherwise  clearly  ex- 
plicable ;  but  it  would  be  out  of  place  here  to  more  than  to  hint 
at  the  importance  of  these  researches ;  to  pursue  them  to  their  fullest 
extent  the  advanced  student  must  have  recourse  to  the  profound 
work  of  Mange,  before  referred  to,  viz.  Application  de  V Analyse 
d.  la  Geomelrie.  We  shall  conclude  the  present  chapter,  with  one  ex- 
ample on  the  determination  of  the  envelope. 

(196.)  The  centre  of  a  sphere  of  given  radius  moves  along  a  given 
plane  curve,  it  is  required  to  determine  the  surface  which  envelopes 
the  sphere  in  every  position. 

Let  the  equation  of  the  given  curve,  along  which  the  centre  moves, 
be 

/3  =  (pa  .  ,  .  .  (1), 
so  that  for  every  abscissa  a  of  this  curve,  the  ordinate  corresponding 
will  be  (pa;  therefore,  the  variable  coordinates  of  the  centre  of  the 
sphere  are  a,  (pa ;  hence  its  equation,  in  any  position,  is 

{x  —  aY  +  (y  —  (pay  +  2"  =-  *^  .   .  .   .   (2), 
hence  the  equations  of  the  characteristic  are, 

{x  —  uf -\-  (y  —  (pay -^  z"  =  r"  }^ 

X  —  a  +  (y  —  (pa  )  (p'a  =  0  j  *  "  *  '   ^  >' 

The  last  equation  is  that  of  a  plane,  passing  through  the  point  (a, 
(pa),  or  centre  of  the  sphere ;  it  is,  moreover,  perpendicular  to  the  tan- 
gent to  the  curve  (1)  at  this  point,  for  the  equation  of  this  tangent  is 

(/3'  —  (3)  =  (p'a  ((pa'  —  (pa), 
and  that  above  is 

t,_<pa  =  — -—  (x  —  a), 
(p  a. 


216  THE  DIFFERENTIAL  CALCULUS. 

SO  that  whatever  be  the  form  of  (p,  the  characteristic  is  always  a  great 
circle  of  the  moveable  sphere,  of  which  the  plane  is  normal  to  the 
curve.  The  species  of  the  curve  which  is  the  characteristic,  being, 
however,  constant,  as  observed  in  art.  (195,)  however  cp  and  a  may 
vary,  the  species  may  be  at  once  determined  by  assuming  a  =  0, 
(pa  =  0,  which  reduces  the  equations  of  the  characteristic  to 

which  belongs  to  a  circle;  the  species,  therefore,  is  a  curve  of  the 
second  order. 

To  determine  the  equation  of  the  envelope,  we  must  eliminate  a 
from  (3),  and  the  resulting  equation  in  x,  y,  z,  will  belong  to  the  en- 
velope ;  thus,  if  the  curve  (1)  be  a  circle  of  radius  a,  then 

—  a 

(pa  =  y/a^  —  a^  .*.  cp'a  =  —  ' 

y/a^  —  a^ 

substituting  these  values  in  the  equations  (3),  they  become 


{x  — a)^  -{■  {y —  's/a^  —  a^)    +  2^  =  r^ 


ay  =^  X  y/  0? — •  a?, 
and  determining,  from  this  last  equation,  the  expression  for  a,  and 
substituting  it  in  the  preceding,  we  shall  obtain,  finally, 


(a  ±  y^  x"  +  f)   =^  r"  —  z", 
for  the  equation  of  the  envelope. 


CaAPTEXt    VI. 

ON  CURVES  OF  DOUBLE  CURVATURE. 

(197.)  In  the  preHminary  chapter  to  the  present  section,  we  inves- 
tigated the  expressions  for  the  tangent  lines  and  normal  planes  to  these 
curves  ;  we  shall  now  discuss  their  general  theory.  As,  however,  in 
the  course  of  this  discussion,  we  shall  sometimes  have  occasion  to 
employ  the  differential  expression  for  the  arc  of  a  curve  of  double 


•THE  DIFFERENTIAL  CALCULUiS.  217 

curvature,  we  shall  commence  by  seeking  the  form  of  this  expres- 
sion. 

(198.)  We  know  that  the  projecting  surface  of  every  curve  of  dou- 
ble curvature,  is  a  cylindrical  surface,  {see^nal.  Geom.)  if,  therefore, 
this  cylindrical  surface  be  developed,  the  curve  will  become  plane, 
and  its  length  will  be  unaltered,  and  the  curvilinear  base  of  the  project- 
ing cylinder,  which  we  shall  here  suppose  to  be  vertical,  will  become 
a  straight  line  on  the  plane  of  xy ;  hence,  for  the  plane  curve  referred 
to  this  straight  line  t,  and  the  axis  of  2,  we  shall  have  (86)  the  ex- 
pression 

{dsf  =  {dzf  4-  {dtf, 
but  t  being  itself  in  reality  the  arc  of  a  plane  curve,  we  have 

{dty  =  {dxf  +  {dyY, 
hence,  by  substitution, 

{dsf  =  {dxf  +  {dyY  +  (c^2)^ 

which  is  the  differential  expression  required. 

Osculation  of  Curves  of  Double  Curvature. 

(199.)  Let 

y  =fx,  z  =  Fx  .  .  .  .  (1), 
and 

Y  =  ■^x,Z  =  Yx  ,  .  .  .  (2), 

be  the  equations  of  two  curves  of  double  curvature,  or  rather  of  the 
projections  of  these  curves  on  the  planes  of  xi/,  xz  :  then,  if  we  con- 
sider the  constants  a,  6,  c,  &c.  which  enter  the  first  pair  of  equations 
as  known,  and  the  constants  A,  B,  C,  &c.  belonging  to  the  second 
pair  as  arbitrary,  these  latter  may  be  determined  so  that  the  curve  to 
which  they  belong  may  touch  the  proposed  or  fixed  curve  (1),  in  any 
given  point,  more  intimately  than  any  other  curve  of  the  family  (2). 
For,  giving  to  x  any  increment,  h,  we  have,  by  Taylor's  theorem, 
.    dv  ,    .    dry      h?      ,    . 

dx  da?    1*2 

and  it  has  been  shown,  (87),  that  if  the  constants  which  enter  the  first 
28 


218  THE  DIFFERENTIAL  CALCULtJS. 

of  the  equations  (2),  be  determined,  all  of  them  from  the  conditions 

,,  dt,       dY     dhi       cFY    , 

the  projection  of  the  curve  (2),  on  the  plane  o^xij,  shall  touch,  more 
intimately,  the  projection  of  (1)  on  that  plane,  than  the  projection  of 
any  other  curve  of  the  family  (2). 

In  like  manner,  if  the  constants  which  enter  the  second  of  the  equa- 
tions (2),  be  all  of  them  determined  from  the  conditions 

„  dz dZ      d~z  __  d'X 

^  ~  ^'d'x  ~~dx'~d?~'  d?'^"" ^^^' 

the  projection  of  the  curve  (2)  on  the  plane  of  xz,  will  touch  more  in- 
timately the  projection  of  (1)  on  that  plane,  than  the  projection  of  any 
other  curve  of  the  family  (2). 

It  is  clear,  therefore,  that  if  all  the  constants  in  the  equations  (2), 
be  determined  conformably  to  the  conditions  (3),  (4),  the  curve  (2) 
will  touch  more  intimately  the  curve  (1)  in  space,  than  any  other  curve 
of  the  family  (2),  and  that  the  contact  will  be  the  less  intimate  as  the 
conditions  (3),  (4),  satisfied  by  the  arbitrary  constants,  are  fewer. 

The  conditions  for  simple  contact,  or  contact  of  the  first  order,  are 
evidently 

—  Y       =  7  ^  =  -^   —  =  -^  fti\ 

'  rfa?         dx^  dx        dx    '   '   '  ' 

and,  for  contact  of  the  second  order,  we  must  have  the  two  additional 
conditions 

d'y    _J^    dr~z    _  d'Z 
dx^  dx^  ^   dx^  dx^  ' 

and  so  on. 

(200.)  From  these  principles,  we  may  very  easily  deduce  the 
equations  of  the  tangent  at  any  point  of  a  curve  of  double  curvature. 

Thus  the  equations  of  any  straight  line  in  space,  are 

y  =  Ax+B',  z  =  \'x  +  B'  .  .  .  .  (1), 

and  these  correspond  to  the  equations  (2)  above,  and  as  four  arbi- 
trary constants  enter,  the  conditions  (5)  may  be  fulfilled  by  them ; 
thus,  takmg  the  two  last  conditions,  we  have,  by  accenting  the  varia- 
bles of  the  curve. 


THE  DIFFERENTIAL  CALCULUS.  219 

^  =  A,  —  =  A', 
dx'  '  dx' 

and,  therefore,  the  two  first  require  that 

so  that  the  equations  (1)  of  the  tangent,  through  any  point  {x',  y',  2'), 


dii' 


(2). 


PROBLEM    I. 

(201.)  To  determine  the  osculating  circle,  at  any  point  in  a  curve 
of  double  curvature. 

In  finding  the  osculating  circle,  at  any  point  of  a  plane  curve,  we 
had  of  course,  the  plane  of  that  curve  given,  but,  in  the  present  case, 
we  have  to  determine  both  the  plane  of  the  circle,  and  its  radius. 
Now  let  us  suppose  that  r  is  the  radius  of  the  osculating  circle,  and 
«»  /3,  7,  are  the  coordinates  of  its  centre,  then  it  is  plain,  that  the 
circle  will  be  a  great  circle  of  the  sphere  whose  equation  is 

{x~ay+{y-^y+  (z  —  yY  =  r^ (1), 

and  since  the  plane  of  the  circle  passes  through  the  point  (a,  /3, 7,)  its 
equation  must  be  of  the  form 

X  —  a  -{-  m{y  —  ^)  +  n{z  —  c)  =  0  .  .   .  .   (2). 
These  two  equations,  combined  with  those  of  the  proposed  curve, 
give  the  values  of  x,  y,  z,  common  to  all,  and  therefore  belong  to  the 
point  where  the  circle  (1),  (2),  meets  the  curve.     We  have,  therefore, 
to  differentiate  the  equations  (1),  (2),  successively,  and  to  consider, 
agreeably  to  the  conditions  (3),  (4),  art.  (199),  that  the  resulting  dif- 
ferential coefficients  belong  as  well  to  the  proposed  curve  at  the 
point,  as  to  this  circle.     For  contact  of  the  first  order  we  have 
(a:_a)  +  {y  —  ^)p   +  (z  —  c)q'^0  ....   (3), 
1  +  mp'  +  nq'  =  0  .  .  .   .   (4), 
and  for  contact  of  the  second  order  we  have,  in  addition. 


220  THr    DITFERENTIAIi    CALCULUS. 

l+l>'^+9'^  +  p"(y-^)+9"(z-7)=0 (5), 

mp"  +  nq"  =  0  .  .  .  .  (6). 
All  these  six  conditions,  therefore,  must  exist  for  the  contact  at  the 
point  {x,  y,  2,)  in  the  proposed  curve  to  be  of  the  second  order ;  and 
as  the  equations  (1),  (2),  of  the  touching  curve,  contain  six  disposable 
constants,  viz.  a,  f3,  y,  r,  m,  n,  all  these  conditions  may  be  fulfilled, 
but  no  more  ;  hence,  the  circle,  determined  agreeably  to  these  con- 
ditions,'will  touch  the  proposed  curve  more  intimately  than  any  other, 
that  is,  it  will  be  the  osculating  circle.  From  equations  (4)  and  (6) 
we  get 

(/'  p" 

m  = ,  »  = ~ , 

q'p"  —  p'  q"  q'p"  —  p'q" 

hence,  equation  (2)  becomes 

x  —  a  +  ^-^ —  {y  —  13)  —  -^^ —  (^z  —  y)=  0, 

IP  —pq  qp  —pq 

or 

^-y-^'^"T^"^"(^-a)+^(y-/3) (7) 

hence,  the  three  conditions  (2),  (4),  (6),  determine  the  plane  of  the 
osculating  circle,  and  which  is  called  the  osculating  plane,  through 
the  proposed  point  {x,  y,  z.)  Equation  (7)  then  represents  this 
plane. 

For  the  coordinates  of  the  centre  of  the  osculating  circle  we  have^ 
from  equations  (1),  (2),  (3), 

(np  —  mq')  r  „  (n  —  o')  r 

^-"^     M     ^y-^  = M—^ 

_  {m  —  p')r 

where  M  is  put  for  the  expression 

x/  \  {np  —  mq'f  +  (w  —  qf  +  (m  —  p'Y\. 
Substituting  these  values  in  (5)  we  have,  for  the  radius  of  the  oscu- 
lating circle, 

(1  -t-  p'^  +  9"=)  M 
{n  —  q')p"—{m—p)q;'' 
Hence,  putting  for  m  and  n  the  values  already  deduced,  and  restoring 
the  value  of  M,  we  have 


THE  DIFFERENTIAL  CALCULUS.  221 

{i+P^  +  q-'r 


V  \p"^  +  g  "'  +  {p'q   —  qfYY 

B  =  v4-  (1  +  p'^  4-  r)  \f—p'  {p'p"  +  q'q") \ 

y  =  -  +  (!+?'  +  g'')  |g"  —  q  (p'p"  +  q'q")  \ 
p"'  +  q"+  ip'q'-q'p'r 

(202.)  The  expression  for  r  may  be  rendered  more  general,  by- 
considering  the  independent  variable  as  arbitrary ;  in  which  case  we 
have  (66), 

„  ^  {dry)  {dx)  —  {(Px)  (dtj)      „  _  jdFz)  (dx)  —  (fe)  (dz) 
^  {d^^  '  ^  {dxf  • 

Also  (198) 

{dxY       ^^P    ^  '^  ' 
hence,  making  these  substitutions  in  the  above  expression,  we  have 

((?5)» 

*""  V \  (dx) {dh))-(ily)(a^)\'+(dz)(a-^x)-{dx) {a^z)\'+{dy) («^2)- (rfs) (ct^y) f  | 

(203.)  If  it  were  required  to  determine  the  circle  having  contact 
of  the  first  order,  merely  with  the  proposed  curve,  only  the  conditions 
(1),  (2),  (3),  (4),  must  be  satisfied;  the  conditions  (2),. (4),  deter- 
mine the  plane  of  this  circle,  that  is  the  tangent  plane,  but  as  the 
condition  (4)  leaves  one  of  the  constants  m,  n,  arbitrary,  the  tangent 
plane  is  not  fixed,  but  may  take  an  infinite  variety  of  positions ;  but 
as  it  must  necessarily  pass  through  the  linear  tangent,  which  is  fixed, 
it  follows  that  a  plane  through  this,  and  revolving  round  it,  is  a  tan- 
gent plane  in  every  position,  in  one  of  which  it  touches  the  curve 
with  a  contact  of  the  second  order,  and  thus  becomes  the  osculating 
plane. 

(204.)  There  is  another  method  of  determining  the  equation  of 
the  osculating  plane,  very  generally  employed  by  French  authors ; 
they  consider  a  curve  of  double  curvature  to  have,  at  every  point,  two 
consecutive  elements,  or  infinitely  small  contiguous  arcs  in  the  same 
plane,  but  not  more,  the  plane  of  these  elements  being  the  osculating 


222  THE  DIFFEREJfTIAL  CALCULUS. 

plane  at  the  point.     The  process,  then,  is  to  assume  the  equation  of 
a  plane  through  the  point 

x  —  x'-\-m{y—y')-^n{z  —  z')  =  0....  (1), 
and  to  subject  it  to  the  condition  of  passing  also  through  the  points 

{x  +  dx',  y'  +  dy',  z'  +  dz% 
and 

X  +  Idx'  +  d\v,  y'  +  2%'  +  dhj ,  z'  +  2dz'  +  d^z'. 
Such  a  process,  the  student  will  at  once  perceive  to  be  exceedingly 
exceptionable ;  for  besides  the  vague  notion  attached  to  the  infinitely 
small  consecutive  arcs,  the  expressions  x  +  dx,  y  +  dy,  and  the  like, 
mean  no  more  in  the  language  of  the  differential  calculus,  than  x,  y, 
&c.,  for  dx,  dy,  &c.  are  not  infinitely  small,  but  absolutely  0,  as  we 
have  all  along  been  careful  to  impress  on  the  mind  of  the  student. 
The  process  is,  however,  susceptible  of  improvement  thus  :  suppose 
the  plane  (1)  passing  through  one  point  {x',y',z')  of  the  curve  passes 
also  through  a  second  point,  of  which  the  abscissa  is  a?'  +  A  x',  where 
Ax'  means  the  increment  of  x,  then  substituting  x  -\-  A  x'  for  x',  the 
equation  (1)  becomes 

X  —  x  ■\-  m  {tj  —  y)  +  n  {z  —  z')  —  (Aa?'  +  inAy  +  wAs') 
=  0   .   .   (2), 
which,  in  virtue  of  (1),  is  the  same  as 

Ax'  -j-  mAy'  +  nAz'  =  0, 

or 

Ay'    .      Az 

Suppose  now  that  these  two  points  merge  into  one,  that  is,  let 
Ax'  =  0,  then 

•        >  +  "'|-  +  »^  =  ----(«); 

hence  the  plane  becomes  determinable  by  the  conditions  (1),  (3). 

Again,  let  this  plane  pass  through  a  third  point,  x  +  Ax',  then  sub- 
stituting this  for  x  in  both  the  equations  (1),  (3),  they  will  furnish  the 
additional  condition 

dy'     ,  dz' 


THE'  DIFFERENTIAL    CALCULUS.  223 

hence,  dividing  by  Ax',  and  supposing  this  third  point  to  coincide  with 
the  former,  that  is,  supposing  Ax'  =  0,  we  have  the  new  condition 

The  equations  (3)  and  (4),  determine  m  and  n,  and  thence  the 
plane  (1 ),  which  is  such  as  to  pass  through  but  one  point  of  the  curve, 
and  at  the  same  time  to  be  so  placed  that  the  most  minute  variation 
from  this  position  will  cause  it  to  pass  through  three  points  of  the 
curve. 

(205.)  By  whatever  process  the  osculating  plane  is  determined, 
the  radius  of  the  osculating  circle  may  be  easily  found  from  consider- 
ations different  from  those  at  (201).  For,  as  the  linear  tangent  to 
the  curve,  must  also  be  tangent  to  the  osculating  circle,  it  follows 
that  the  centre  of  this  circle  must  be  on  the  normal  plane,  as  well  as 
on  the  osculating  plane ;  it  must,  therefore,  lie  in  the  line  of  intersec- 
tion of  this  normal  plane,  with  its  consecutive  normal  plane ;  hence, 
if  this  line  be  determined,  the  combination  of  its  equation  with  that 
of  the  osculating  plane,  will  give  the  point  sought.  Now  (189)  the 
line  of  intersection  of  consecutive  normal  planes  is 

x  —  x'-^-p'  {y—y')+q'  {z  —  z')  =  0  \ 

p"  iy  —  y')  +  q"  i^  —  ^')  —p"  —  q"-i  =  oi 

therefore,  the  centre  is  to  be  determined  by  combining  these  equa- 
tions with  that  of  the  osculating  plane,  viz. 

being  precisely  the  samfe  equations  as  those  employed  before,  for  the 
same  purpose.  If  the  origin  be  at  the  point,  and  the  tangent  be  the 
axis  of  ar,  then  x',  y',  z',p',  q',  are  each  0 ;  therefore,  the  equations  of 
the  line  of  intersection  are 

q"      .    1 

X  =  0,y  =  —2-z    +— , 

and  the  equation  of  the  osculating  plane 

p"z-q"y  =  0; 

this,  therefore,  is  perpendicular  to  the  line  of  intersection.     {AncU. 
Geom.) 
(206.)  The  expressions  in  (201)  for  the  coordinates  of  the  centre 


224  THE    DIFFERENTIAL    CALCULUS. 

of  the  osculating  circle  will  become  very  simple  by  introducing  the 
substitutions  furnished  by  art.  (202);  the  results  of  these  substitutions 
will  be 


the  independent  variable  being  s.     (See  JVote  D.) 

PROBLEM  II. 

(207.)  To  determine  the  centre  and  radius  of  spherical  curvature 
at°any  point  in  a  curve  of  double  curvature. 

We  are  here  required  to  determine  a  sphere  in  contact  with  the 
proposed  curve  at  a  given  point,  such  that  a  line  on  its  surface  in  the 
direction  of  the  proposed  may  in  the  vicinity  of  the  point  be  closer  to 
the  curve  than  if  any  other  sphere  were  employed.  In  the  direction 
of  the  curve  the  z  and  the  y  of  the  sphere  must  be  both  functions  of 
X,  so  that  the  equation  of  the  sphere  is  resolvable  into  two,  corres- 
ponding to  the  equations  (2)  art.  (199),  which  two  equations  belong 
to  the  curve  which  osculates  the  proposed.  The  actual  resolution  of 
the  equation  into  two  is  obviously  unnecessary ;  it  will  be  sufficient 
in  that  equation  to  consider  x  as  the  only  independent  variable. 

The  general  equation  of  a  sphere  is 

{x  —  ay+  {y—^y-  +  {z  —  yy  =  r'  ....   (1), 
and  the  particular  sphere  required  will  be  that  whose  constants  are 
determined  from  the  following  differential  equations  : 

x  —  a  +  p'{y  —  l3)  +  q'{z  —  y)  =  0   .   .   .   .   (2) 

P"  {y-(3)  +  q"  {z-y)  +  l-\-p"+q"  =  0 (3) 

p'"  (!/  -  /3)  +  q'"  (^  -  7)  +  3  W  +  <lY)  =0 (4). 

These  four  equations  fix  the  values  of  the  parameters  a,  /3,  y,  r^ 
and,  therefore,  determine  both  the  position  and  magnitude  of  the  os- 
culating sphere.  If  the  origin  of  coordinates  be  at  the  proposed 
point,  and  the  linear  tangent  be  taken  for  the  axis  of  x,  the  determina- 
tion becomes  easy,  for  x,  y,  z,  being  each  =  0,  as  also  p',  q',  the  fore- 
going equations  (2),  (3),  (4),  become 

a  =  0,|>"/3  +  q"y  =  1,  p'"/3  +  q"'y  =  0, 

_        q"  _       p'" 

*'•  ^  ~  p"q"'  —f'q'"'^  ~  q"p"'  —  q''p"  ' 


TttE  DIFFERENTIAL  CALCULVll.  225 

iience,  by  substitution  in  (1), 


r 


^/  p"'a  +  q'"- 


p"q"'-9"p"' 

(208.)  We  already  know  that  if  to  every  point  in  a  curve  of  double 
curvature  normal  planes  be  drawn,  the  intersections  of  these  planes 
with  the  consecutive  normal  planes  will  be  the  characteristics  of  the 
developable  surface  which  they  generate,  and  the  intersection  of  any 
characteristic  with  the  consecutive  characteristic  will  be  a  point  in 
the  edge  of  regression,  corresponding  to  the  given  point  on  the  pro- 
posed curve.  Now  equation  (2)  above  being  that  of  the  normal 
plane,  this  point  is  determined  by  precisely  the  same  equations  (2), 
(3),  (4),  as  determine  the  centre  of  spherical  curvature,  these  points, 
therefore,  are  one  and  the  same,  as  might  be  expected ;  hence  the 
locus  of  the  centres  of  spherical  curvature  forms  the  edge  of  regres- 
sion of  the  developable  surface  generated  by  the  intei  sections  of  the 
consecutive  normals.  If  then  by  means  of  one  of  the  equations  of 
the  proposed  curve  and  the  three  equations  of  condition  mentioned 
we  eliminate  x,  y,  z,  and  then  perform  the  same  elimination  by  means 
of  the  other  equation  of  the  curve  and  the  same  conditions,  we  shall 
obtain  two  resulting  equations  in  a,  ^,  y,  which  will  be  the  equations 
of  the  edge  of  regression. 

PROBLEM  III. 

(209.)  To  determine  the  points  of  inflexion  in  a  curve  of  double 
curvature. 

Since  a  curve  of  double  curvature  as  its  name  implies  has  curva- 
ture in  two  directions,  if  at  any  point  its  curvature  in  one  direction 
changes  from  concave  to  convex  the  point  is  called  a  point  of  simple 
inflexion.  But  if  at  the  same  point  there  is  also  a  like  change  of 
curvature  in  the  other  direction,  the  point  is  then  said  to  be  one  of 
double  inflexion.  In  other  words,  if  but  one  projection  of  the  tan- 
gent crosses  the  projected  curve  the  point  is  one  of  simple  inflexion, 
but  if  the  tangent  cross  the  curve  in  both  projections  then  the  point  is 
one  of  double  inflexion.  As  in  plane  curves  the  teuigent  line  has 
contact  one  degree  higher  at  a  point  of  inflexion,  so  here  the  contact 
of  the  osculating  plane  is  one  degree  higher.  Hence,  at  such  a 
point  besides  the  conditions  in  (201)  which  fix  the  osculating  plane, 

29 


226  THE  DIFFEREKTIAL  CALCULUS. 

we  must  at  a  point  of  simple  inflexion  have  the  additional  condition 
arising  from  differentiating  (6),  viz. 

mp'"  +  nq"  =  0. 

Eliminating  —  from  this  and  equation  (6)  vi^e  have 

p"q"'  -  q"p"% 
which  condition  renders  the  expression  for  the  radius  of  spherical 
curvature  at  the  point  infinite,  as  it  ought.*  Unless,  therefore,  this 
condition  exist,  the  point  cannot  be  one  of  inflexion ;  but  the  point 
for  which  the  condition  holds  may  be  one  of  inflexion,  yet  to  deter- 
mine this  the  curve  must  be  examined  in  the  vicinity  of  the  point. 

As  to  points  of  double  inflection,,  it  is  evident  from  what  has 
been  said  (121)  with  respect  to  plane  curves  that  such  points  must 
fulfil  the  conditions 

p"  =  0  or  CO  ,  g"  =  0  or  CO  , 
and  these  render  the  radius  r  of  absolute  curvature  infinite  or  0. 

Evolutes  of  Curves  of  Double  Curvature. 

(210.)  In  speaking  of  the  evolutes  of  plane  curves  we  observed 
(103,)  that  the  evolute  of  any  plane  curve  was  such  that  if  a  string 

*  The  French  mathematicians  consider  a  point  of  simple  inflexion  to  be  that  at 
which  three  consecutive  elements  of  the  curve  lie  in  the  same  plane.  In  a  recent 
publication  from  the  university  of  Cambridge  the  author  has  attempted  to  deduce 
the  above  equation  of  condition,  by  viewing  the  point  of  inflexion  after  the  manner 
of  the  French.  He  has  however  confounded  the  consecutive  elements  of  a  curve 
with  what  the  same  writers  term  consecutive  points ;  moreover,  after  having  estab- 
lished the  conditions  necessaiy  for  the  plane 

z  =  Aa;  +  Bj/  +  C, 
passing  through  one  point  (r,  j/,  r,)  in  the  curve,  to  pass  also  through  two  points 
consecutive  to  this,  viz.  the  conditions 

^=A  +  B^.|^=A+Bf?i 
dx  ax  '  ax  ax 

where  y„  z„  belong  to  one  of  tlie  consecutive  points,  it  is  inferred  that 

dx^  dx^'    dx^  dx^ 

an  inference  which  is  quite  unwarrantable,  and  which  cannot  exist  unless  the  plane 
pass  through/owr  consecutive  points  instead  of  three. 


THE    DIFFERENTIAL   CALCULUS.  227 

were  wrapped  round  it  and  continued  in  the  direction  of  its  tangent 
till  it  reached  a  point  in  the  involute  curve,  the  unwinding  of  this  string 
would  cause  its  extremity  to  describe  the  involute.  But  besides  the 
plane  evolute  hitherto  considered,  there  are  numberless  curves  of 
double  curvature  round  which  the  string  might  be  wound  and  con- 
tinued in  the  direction  of  a  tangent  till  it  reached  the  involute,  which 
would  equally,  by  unwinding,  describe  this  involute  ;  and  generally 
every  curve,  whether  plane  or  of  double  curvature,  has  an  infinite 
number  of  evolutes,  as  we  are  about  now  to  show. 

(211.)  If  through  the  centre  of  a  circle,  and  perpendicular  to  its 
plane,  an  indefinite  straight  line  be  drawn,  and  any  point  whatever 
be  taken  in  this  line,  then  it  is  obvious  that  this  point  will  be  equally 
distant  from  every  point  in  the  circumference  of  the  circle,  so  that,  if 
a  line  be  drawn  from  it  to  the  circumference,  this  line,  in  revolving 
round  the  perpendicular  under  the  same  angle,  will  describe  the  cir- 
cumference. Such  a  point  is  called  a  pole  of  the  circle,  so  that  every 
circle  has  an  infinite  number  of  poles,  the  locus  of  which  is  determined 
when  the  places  of  any  two  are  given. 

(212.)  Now,  as  respects  curves  of  double  curvature,  we  have  seen 
that  the  centre  of  the  circle  of  absolute  curvature  corresponding  to 
any  point  is  in  the  line  where  the  normal  at  this  point  is  intersected  by 
its  consecutive  normal,  the  centre  itself  being  that  point  in  this  line 
where  it  pierces  the  osculating  plane,  which  (205)  is  the  plane  drawn 
through  the  tangent  line  perpendicular  to  this  line  of  intersection,  or 
characteristic  ;  hence  the  characteristic  corresponding  to  any  point 
in  the  curve  is  the  locus  of  the  poles  of  curvature  at  that  point,  and 
the  intersection  of  this  characteristic,  with  the  perpendicular  to  it  from 
the  corresponding  point  of  the  curve,  is  that  particular  pole  which  is 
the  centre  of  absolute  curvature,  the  perpendicular  itself  being  the 
radius. 

As  the  locus  of  the  poles  corresponding  to  any  point  is  no  other 
than  the  characteristic,  the  locus  of  all  the  poles  corresponding  to  all 
the  points  of  the  curve  must  be  the  locus  of  all  the  characteristics, 
and  therefore  (190)  a  developable  surface. 

(213.)  Suppose  now  through  any  point,  P,  of  the  curve  a  normal 
plane  is  drawn  of  indefinite  extent,  the  characteristic  or  line  of  polef> 
corresponding  to  the  point  will  be  in  this  plane  ;  let,  therefore,  any 
straight  line  be  drawn  from  P  to  intersect  this  line  of  poles  in  the  point 


228  THE  DIFFERENTIAL  CALCULUS. 

Q,  and  be  continued  indefinitely.  If  this  normal  plane  be  conceived 
to  move,  so  that,  while  P  describes  the  proposed  curve,  the  plane 
continues  to  be  normal,  the  characteristic  will  undergo  a  correspond- 
ing motion,  and  will  generate  the  developable  surface  corresponding 
to  the  curve  described  by  P,  and  this  motion  of  the  characteristic  will 
cause  a  corresponding  motion  of  the  point  Q,  not  only  in  space,  but 
along  the  arbitrary  line  from  P,  which  has  no  motion  in  the  moving 
plane.  As,  therefore,  Q  moves  along  the  characteristic  successive 
portions  QQ'  of  the  line,  PQ  will  apply  themselves  to  the  surface 
which  the  moveable  characteristic  generates,  and  there  form  a  curve 
to  which  always  the  unapplied  portion  QP  is  a  tangent.  Now  the 
normal  plane  being  in  every  position  tangent  to  the  surface  through- 
out the  whole  length  of  the  characteristic,  it  is  obvious  that,  in  the 
above  generation  of  this  surface,  nothing  more  in  effect  has  been  done 
than  the  bending  of  the  original  normal  plane,  supposed  flexible,  into 
a  developable  surface.  If,  therefore,  we  now  perform  the  reverse 
operation,  that  is,  if  we  unbend  the  normal  plane,  the  point  P  will  de- 
scribe the  curve  of  double  curvature,  and  the  curve  QQ'  traced  on 
the  developable  surface  will  become  the  straight  line  PQ ;  so  that  the 
curve  of  double  curvature  may  be  described  by  the  unwinding  of  a 
string  wrapped  about  the  curve  Q'Q,  and  continued  in  the  direction 
QP  of  its  tangent,  till  it  reaches  the  point  P  in  the  proposed  curve. 
It  follows,  therefore,  that  the  curve  Q'Q  is  an  evolute  of  the  curve  of 
double  curvature  proposed,  and,  moreover,  that,  as  the  line  PQ  ori- 
ginally drawn  was  quite  arbitrary,  the  proposed  curve  has  an  infinite 
number  of  evohdes  situated  on  the  developable  surface,  which  is  the 
locus  of  the  poles  of  the  proposed ;  hence  the  locus  of  the  poles  is  the 
locus  of  the  evolutes. 

If  the  original  line  PQ  be  perpendicular  to  the  corresponding  line 
of  poles  or  characteristic,  then,  since  this  characteristic  moves  in  the 
moving  plane  while  PQ  remains  fixed,  PQ  cannot  continue  to  be  per- 
pendicular to  the  characteristic ;  but  the  radius  of  absolute  curvature 
is  always  perpendicular  to  the  characteristic,  this  radius  therefore 
cannot  continue  to  intersect  the  characteristic  in  the  point  Q,  so  that 
the  locus  of  the  centres  of  absolute  curvature  is  not  one  of  the  evolutes 
of  the  proposed  curve, 

(214.)  Should  the  curve  which  we  have  all  along  considered  of 
double  curvature  be  plane,  then,  indeed,  since  the  characteristics  are 


THE  DIFFERENTIAL  CALCULUS.  229 

^11  parallel,  and  perpendicular  to  the  plane  of  the  cun'e,  the  line  PQ 
once  perpendicular  will  be  always  perpendicular  to  the  chajacteristic, 
so  that  then  Q  will  coincide  with  the  centre  of  curvature,  PQ  being 
no  other  than  the  radius  of  curvature,  the  locus  of  the  centres  being 
the  plane  evolute  before  considered.  But  when  PQ  is  not  drawn  per- 
pendicular to  the  original  characteristic,  but  is  inclined  to  it  at  an  an- 
gle a,  then  it  always  preserves  this  inclination  during  the  generation 
of  the  cylindrical  surface  which  is  the  locus  of  the  poles,^  therefore 
every  curvilinear  evolute  of  a  plane  curve  is  a  helix  described  on  the 
surface  of  the  cyUnder,  which  is  the  locus  of  the  poles  of  the  plane 
curve. 

Every  curve  traced  on  the  surface  of  a  sphere,  has,  for  the  locus  of 
its  evolutes,  a  conical  surface  whose  vertex  is  at  the  centre  of  the 
sphere  ;  because  the  normal  planes  to  the  curve  being  also  normal 
planes  to  the  spheric  surface,  all  pass  through  the  centre. 

(215.)  From  what  has  now  been  said,  it  is  obvious  that  if  from  any 
point  in  a  curve  a  line  be  drawn  to  touch  the  developable  surface  which 
is  the  locus  of  its  poles,  and  its  prolongation  be  wound  about  the  sur- 
face without  twisting,*  it  will  trace  one  of  the  evolutes,  and,  as  the 
string  may  be  drawn  to  touch  the  surface  in  every  possible  direction, 
it  follows  that  every  developable  line  on  the  surface  will  be  an  evo- 
lute. If  the  curve  be  plane,  the  evolutes  are  all  on  the  cylindrical 
surface  whose  base  is  the  plane  evolute. 

As  obviously  a  developable  line  is  the  shortest  on  the  surface  that 
can  join  its  extremities,  it  follows  that  the  shortest  distance  between 
two  points  of  an  evolute  measured  on  the  surface  is  the  arc  of  that 
evolute  between  them. 

PROBLEM    IV. 

(216.)  Having  given  the  equations  of  a  curve  of  double  curvature 
to  determine  those  of  any  one  of  its  evolutes. 

All  the  evolutes  of  the  curve  being  on  the  same  developable  sur- 

*  This  is  what  I  understand  Nonge  to  mean,  when  he  says  {Jlpp.  de  VAncd.  de 
Gdom.  p.  348,)  "si  I'on  plie  librement  sur  cette  surface  le  prolongement  de  cette 
tangente."  It  seems  not  improper  to  call  such  lines  placed  on  a  developable  sur- 
face developable  lines,  and  those  which  form  curves  on  the  developed  surface  hoist. 
ed  lines.  Of  these  two  species  of  lines  all  the  former  are  evolutes,  but  none  of  the 
latter  are. 


230  THE  DIFFERENTIAL  CALCULUS. 

face,  the  equation  of  this  surface  must  be  common  to  them  all,  and 
we  have  already  seen  (194)  how  the  equation  of  the  surface  is  to  be 
determined,  so  that  it  only  remains  to  find  for  each  evolute  a  particu- 
lar equation  which  distinguishes  it  from  all  the  others,  and  determines 
its  course  on  the  developable  surface.  In  order  to  this  let  us  consi- 
der that  each  evolute  must  be  such  that  the  prolongation  of  its  tan- 
gent at  any  point  always  cuts  the  involute,  or,  which  is  the  same 
thing,  the  tangent  to  the  projection  of  the  evolute  at  any  point  passes 
through  the  corresponding  point  in  the  projection  of  the  evolute ; 
therefore,  considering  the  plane  of  xy  as  that  of  projection,  we  have, 
for  the  tangent  at  any  point  {x',  y')  in  the  projected  evolute, 

and,  since  the  same  line  passes  through  a  point  {x,  y,)  in  the  project- 
ed involute,  its  equation  is  also 

Y  —  y'  =  yLz:yfx  —  x') 
,.,M.  -v'—y. 

dx'  x'  —  ar ' 
hence,  combining  this  equation  with  that  of  the  developable  surface, 
determined  agreeably  to  the  process  pointed  out  in  article,  194,  and 
eliminating  x,  y  being  a  given  function  of  x^  we  shall  have  two  equa- 
tions in  x\  y',  z',  of  which  one  will  contain  partial  differential  coeffi- 
cients of  the  first  order,  and  which  together  will  represent  all  the  evo- 
lutes.  To  find  that  particular  one  which  is  fixed  by  any  proposed 
condition,  it  will  be  necessary  to  discover,  by  the  aid  of  the  integral 
calculus,  the  primitive  equation  from  which  the  differential  equation 
mentioned  is  deducible  ;  this  primitive  equation  will  involve  an  arbi- 
trary constant,  whose  value  may  be  fixed  by  the  proposed  condition, 
and  thus  the  equations  of  the  particular  evolute  will  be  determined. 

We  shall  terminate  this  section  by  subjoining  a  few  miscellaneous 
propositions. 


THE  DIFFERENTIAL  CALCULUS.  231 


CHAFTER  VII. 
MISCELLANEOUS  PROPOSITIONS. 

PROPOSITION  I. 

(217.)  To  prove  that  the  locus  of  all  the  linear  tangents  at  any 
point  of  a  curve  surface  is  necessarily  a  plane. 

This  property  we  have  hitherto  assumed ;  it  may,  however,  be  de- 
monstrated as  follows : 

Let  the  equation  of  any  curve  surface  be 

«  =/(^'2/)  •  •  •  •  (1)' 
X  and  y  being  the  independent  variables. 

Through  any  given  point  on  this  surface  let  any  curve  be  traced, 
then,  the  projection  of  this  curve  on  the  plane  of  xy  will  be  represented 
by 

y  =  (px  .  .  .  .  (2), 

which  will  equally  represent  the  projecting  cylinder ;  hence  the  com- 
bination of  the  equations  (1),  (2),  completely  determines  the  curve, 
and  its  projection  on  the  plane  of  a;^  may  be  found  by  eliminating  y 
from  these  equations ;  the  result  of  this  elimination  will  be  the  equa- 
tion 

z=f{x,C!)x)=-^x....   (3), 

therefore,  since  the  linear  tangent  in  space  is  projected  into  tangents 
to  these  two  curves  (2),  (3),  its  equations  must  be 


—  X)  \ 

—  x')) 


where  a;',  ?/',  2',  are  the  coordinates  of  the  proposed  point  on  the  sur- 

face.    Now  —7—  is  the  total  differential  coefficient  derived  from  the 
ax 

function  s  =  /  (a?,  t/),  in  which  y  is  considered  as  a  function  of  x given 

by  the  equation  (2),  that  is 


233  TUB  DIFFERBNTIAI.  CALCULUS. 

d^x  _  cdz^   __    ^  ^    ,  d(px 

ITx       te«  ~^   '  ^  dT' 

hence,  by  substitution,  the  equations  of  the  tangent  in  space  become' 

y—y  --rfT"^''— "^^i 

.-.'  =  ip'  +  ,''£)ix-x')]""^'^' 

Now,  to  obtain  the  locus  of  the  tangents  whatever  be  the  curve  through 
the  point  {x',  tj',  z'),  we  must  eliminate  the  function  (par,  on  which  alone 
the  nature  of  the  curve  depends.  Executing  then  this  elimination  by 
means  of  the  equations  (4)  and  there  results  for  the  required  locus 
the  equation 

z—z'  =-.p'  (^x  —  x')  +  q'(y  —  y'), 
which  is  that  of  a  plane. 

PROPOSITION  II. 

(218.)  Given  the  algebraic  equation  of  a  curve  surface  to  deter- 
mine whether  or  not  the  surface  has  a  centre. 

That  point  is  called  the  centre  which  bisects  all  the  chords  drawn 
through  it,  so  that  if  the  equation  of  the  surface  is  satisfied  for  any 
constant  values  x\  y\  z',  it  will  equally  be  satisfied  for  the  same  val- 
ues taken  negatively,  that  is,  for  —  x',  —  y',  —  z',  provided  the  ori- 
gin of  coordinates  be  placed  at  the  centre,  so  that  if  no  point  exists 
for  the  origin  of  coordinates,  in  reference  to  which  the  equation 

f{x,rj,z,)  =  0 
of  the  surface  remains  the  same  whether  the  signs  of  the  variables  be 
assumed  all  +  or  all  — ,  then  we  may  conclude  also  that  no  centre 
exists. 

The  mode  of  proceeding,  therefore,  is  to  assume  the  indeterminates 
x^,  J/,,  2 ,  for  the  coordinates  of  the  unknown  centre,  and  to  transport 
the  origin  of  the  axes  to  that  point  by  substituting  in  the  equation  of 
the  surface  x  -]-  x^,y  +  y,  z  +  2 ,  for  x,  y,  2.  This  done  we  may 
readily  deduce  equations  of  condition  which  will  give  the  proper  val- 
ues of  X,  j/^,  2,,  if  a  centre  exists,  or  will  show,  by  their  incongruity, 
that  the  surface  has  no  centre.  Thus,  suppose  the  equation  of  the 
surface  is  of  an  even  degree,  then  we  must  equate  to  0  the  coefficients' 


THE  DIFFERENTIAL.  CALCULUS. 

of  all  the  odd  powers  and  combinations  of  x,  y,  z,  since  the  terms  into 
which  these  enter  would  change  signs  when  the  variables  change 
signs  :  we  obtain  in  this  way  the  equations  of  condition.  If  the  equa- 
tion of  the  surface  be  of  an  odd  degree,  then  we  must  equate  to  zero 
the  coefficients  of  all  the  even  powers  and  combinations  of  a*,  ?/,  2 ;  so 
that  only  odd  powers  and  combinations  may  effectively  enter  the 
equation,  for  then  whether  the  variables  be  all  +  or  all  —  the  function 
f(x,  y,  s,)  will  still  be  0. 

Now  the  differential  calculus  furnishes  us  at  once  with  the  means 
of  obtaining  the  several  expressions  which  we  must  equate  to  zero 
without  actually  substituting  x  -{-  x^,y+  y,?  2:  +  z^,  for  x,  y,  z,  in  the 
equation  of  the  surface.  For  if  we  conceive  these  substitutions  made 
in  the  function /(a?, »/,  z),  we  may  consider  the  result  as  arising  from 
^/' !//'  ^/»  taking  the  respective  increments  x,  y,  z,  and  we  know  that 
every  such  function  may  by  Taylor's  theorem  be  developed  accord- 
ing to  the  powers  and  combinations  of  the  increments,  and  that  the 
several  terms  of  the  development  consist  each  of  the  partial  differen- 
tial coefficients  of  the  preceding  term,  the  first  being/ (a:^,  ?/^,  zj. 
Hence,  if  the  coefficients  of  the  first  powers  of  x,  y,  z,  are  to  be  re- 
spectively zero,  then  we  have  to  equate  to  zero  each  of  the  partial 
coefficients  derived  from  u^  =  /(^,»  y,»  «,>)  =  0,  or,  which  is  the  same 
thing,  from  u=f{x^  y,  z,)  =  0  the  proposed  equation  ;  if  the  coeffi- 
cients of  the  second  powers  and  combinations  of  x,  y,  z,  are  to  be  ren- 
dered each  0,  then  we  shall  have  to  equate  to  zero  each  partial  coef- 
ficient derived  from  again  differentiating,  and  so  on. 

As  an  illustration  of  this,  let  the  general  equation  of  surfaces  of  the 
second  order 

Ar'  +  Ay  -\-  A"z'  +  2Byz  +  2B'zx  +  2B"xi/  )  _  ^  _  ,, 

+  2Cx  +  2C'y  +  2C"z  +  E         J  -"-«...  (1) 

be  proposed,  then  the  degree  of  the  equation  being  even,  the  coeffi- 
cients of  the  odd  powers  of  the  variables  in  the  equation  arising  from 
putting  X  -\r  x^,  y  -\r  y^fZ  -\-  z^,  for  x,  y,  z,  are  to  be  equated  to  0,  and 
as  the  equation  is  but  of  the  second  degree,  these  odd  powers  will  be 
of  the  first ;  hence  we  have  merely  to  equate  the  first  partial  differen- 
tial cofficients  to  0,  that  is 

30 


234  THE  DIFFERENTIAL  CALCULUS. 

^  =  Ax   +  B'z  +  B"y  +  C  =  0  ^ 

^  =  A'v  +  Bz  +  B"x  +  C  =  0  V      .  .  .  (2). 
dy 

^  =  M'z  +  B'z  +  B«    +  C"  =  0 
dz  ^ 

The  values  of  x,  y,  2,  deduced  from  these  equations  are  the  coordi- 
nates x^yy^,  2 ,  of  the  centre.     These  values  may  be  represented  by 
_N'       _N'      _N" 
^  -  D '  ^'  ~  D  '  ^'  ~  D  * 

where 

D  =  AB''  4-  AB""  +  A"B"2  —  AAA"  —  2BB'B", 
so  that  the  surface  has  a  centre  if  D  is  not  0,  but  if  D  =  0  and  the 
numerators  all  finite,  the  surface  has  no  centre,  and,  lastly,  if  D  =  0 
and  either  of  the  numerators,  also  0,  then  the  surface  has  an  infinite 
number  of  centres,  and  is,  therefore,  cylindrical. 

The  equations  of  condition  (2)  are  the  same  as  those  at  page 

of  the  Analytical  Geometry. 

PROPOSITION  III. 

(219.)  To  determine  the  equation  of  the  diametral  plane  in  a  sur- 
face of  the  second  order  which  will  be  conjugate  to  a  given  system 
of  parallel  chords. 

Let  the  inclinations  of  the  chords  to  the  axes  be  a,  /3,  y,  then  the 
equations  of  any  one  will  be 

x  =  mz  -]r  p^y  =  nz-\-  q  .  .  .  .  (1), 

where 

cos.  a  cos.  ^ 

m  = ,  n  = . 

cos.  y  cos.  y 

For  the  points  common  to  this  hne  and  the  surface  we  must  combine 

this  equation  with  equation  (1)  last  proposition,  and  we  shall  have  a 

result  of  the  form 

Rz^  +  Ss  -f  T  =  0  .  .  .  .  (2), 

which  equation  will  furnish  the  two  values  of  2  corresponding  to  the 
two  extremities  of  the  diameter,  and  therefore  half  the  sum  of  these 
values  will  be  the  z  of  the  middle,  that  is, 


THE    DIFFERENTIAL    CALCULUS.  [  285 

2  =  — ^.•.  2R2==S  +  0  .  .  .  .  (3), 

which  is  obviously  the  differential  coefficient  derived  from  (2),  or, 
which  is  the  same  thing,  the  total  differential  coefficient  derived  from 
( 1 )  last  proposition,  in  which  x  and  y  are  functions  of  z  given  by  the 
equations  (1).     This  differential  coefficient  is,  therefore, 

dtt.  du  dx       du  dy       du  

dz*        dx  dz       dy  dz       dz 

du   ,       du   ,   du 
=  m—-\-n  —  +  -j-  =  Q,....  (4  , 
dx  dy       dz 

where  p  and  5,  the  only  quantities  which  vary  with  the  chord,  are 
eliminated  ;  hence,  this  last  equation  represents  the  locus  of  the  mid- 
dle points  of  the  chords  or  the  diametral  surface,  and  it  is  obviously 
a  plane. 

By  actually  effecting  the  differentiations  indicated  in  equation  (4) 
upon  the  equation  (1)  last  proposition,  we  have  for  the  equation  of  the 
required  diametral  plane, 

m  (Ax  +  B'z  +  B"y  +  C)  +  n  (A'^  +  Bs  +  B"x  +  C) 
+  A"z  +  Br/  +  Bx  +  C"  =  0, 
or 

(Am  +  B'  +  B"n)  x  +  (A'n  +  B  +  B"m)  y  + 
(A"   +  Bn-I-  B'm)  z  +  Cm  -f  C'n  +  C"  =  0. 

PROPOSITION  IV. 

(220.)  A  straight  line  moves  so  that  three  given  points  in  it  con- 
stantly rest  on  the  same  three  rectangular  planes ;  required  the  sur- 
face which  is  the  locus  of  any  other  point  in  it. 

Let  the  proposed  planes  be  taken  for  those  of  the  coordinates,  and 
let  the  coordinates  of  the  generating  point  be  x,y,  z,  and  the  invaria- 
ble distances  of  this  point  from  the  three  points  resting  on  the  planes 
of  yz,  xz,  and  xy,  X,  Y,  Z.  The  coordinates  of  these  three  points 
will  be 

In  the  plane  of  yz,  0,  y\  z' 

xz,  x",  0,  z" 
xy,x'",y"'0. 


X 

y  —  y'  _ 

Y                Z 

y     -y  —  y'" 

X 

z~z' 

z 

Y                Z 

—  z"             z 

236  THE  DIFFERENTIAL  CALCULUS. 

Then,  since  the  parts  of  any  straight  line  are  proportional  to  their 
projections  on  any  plane,  each  part  having  the  same  inclination  to  it, 
it  follows  that  if  we  project  successively  each  of  the  parts  X,  Y,  Z, 
on  the  three  coordinate  planes,  we  shall  have  the  relations 


.  .  (1). 


X  Y  Z 

But  the  part  X  of  the  moveable  straight  line  comprised  between  the 
generating  point  {x,  y,  z,)  and  the  point  (0,  r/,,  z^,),  resting  on  the  plane 
of  y,  z,  has  for  its  length  the  expression 

or 

J2  X^       ^        X''        •  •  •  •  ^^' 

but  from  the  equations  (1) 

!/  —  !/'_  y     z  —  z'  _  z_ 
X  Y '      X  Z  '  . 

hence,  by  substitution,  (2)  becomes 

\.  ^—  -\-  =1 

X*"      Y^       2^        ' 

consequently,  the  surface  generated  is  always  of  the  second  order. 
The  surface  would  still  be  of  the  second  order  if  the  three  directing 
planes  were  oblique  instead  of  rectangular,as  is  shown  by  JVf.  Dupin, 
in  his  Developpements,  p.  342,  whence  the  above  solution  is  taken. 

PROPOSITION    V. 

(221.)  To  determine  the  line  of  greatest  inclination  through  any 
point  on  a  curve  surface. 

The  property  which  distinguishes  the  line  of  greatest  inclination 
through  any  point  is  this,  viz.  that  at  every  point  of  it  the  linear  tan- 
gent makes  with  the  horizon  a  greater  angle  than  any  other  tangent 
to  the  surface  drawn  through  the  same  point  of  the  curve.  Now,  a» 
all  the  linear  tangents  through  any  point  are  in  the  tangent  plane  ta 


THE  DIFFERENTIAL  CALCULUS.  237 

the  surface  at  that  point,  that  one  which  is  perpendicular  to  the  trace 
of  the  tangent  plane  will  necessarily  be  the  shortest,  and  therefore 
approach  nearest  to  the  perpendicular,  that  is,  it  will  form  a  greater 
angle  with  the  horizon  than  any  of  the  others.  We  have,  therefore^ 
to  determuie  the  curve  to  which  the  linear  tangent  at  every  point  i» 
always  perpendicular  to  the  horizontal  trace  of  the  tangent  plane  ta 
the  surface  through  the  same  point,  or,  which  is  the  same  thing,  the 
projection  of  the  linear  tangent  on  the  plane  ofxy  must  be  perpen- 
dicular to  the  trace  of  the  tangent  plane. 

Now  the  equation  of  the  projection  of  the  linear  tangent  at  any 
point  is 

and,  by  putting  z  =  0  in  the  equation  of  the  tangent  plane,  we  have 
for  the  trace  in  the  plane  of  xy  the  equation 

—  z'=p'{x—x')+q'{y-y'), 
and,  since  these  two  lines  are  to  be  always  perpendicular  to  each 
other,  we  must  have  throughout  the  curve  the  general  condition. 

dx  p' '  dx  " 
p'  and  q'  being  derived  from  the  equation  of  the  surface ;  so  that  the 
values  of  these  being  obtained  in  terms  of  x  and  y,  and  substituted  in 
the  equation  just  deduced,  the  result  will  be  the  general  differential 
equation  belonging  to  the  projection  of  every  curve  of  greatest  inclina- 
tion that  can  be  drawn  on  the  proposed  surface.  To  determine  that 
passing  through  a  particular  point,  or  subject  to  a  particular  condition, 
we  must,  by  help  of  the  integral  calculus,  determine  the  general 
primitive  equation  from  which  the  above  is  deducible,  this  primitive 
will  involve  an  arbitrary  constant  which  may  be  fixed  by  the  proposed 
condition,  and  thus  the  particular  line  be  represented. 


PROPOSITION    VI. 

(222.)  The  six  edges  of  any  irregular  tetraedron  or  triangular 
pyramid  are  opposed  two  by  two,  and  the  nearest  distance  of  two  op- 
posite edges  is  called  breadth;    so  that  the  tetraedron  has  three 


238 


THE  DIFFERENTIAL  CALCULUS. 


breadths  and  four  heights.  It  is  required  to  demonstrate  that  in  every 
tetraedron  the  sum  of  the  reciprocals  of  the  squares  of  the  breadths  is 
equal  to  the  sura  of  the  reciprocals  of  the  squares  of  the  heights. 

Let  the  vertex  of  the  tetraedron  be  taken  for  the  origin  of  the  rec- 
tangular coordinates,  and  let  also  one  of  the  faces  coincide  with  the 
plane  of  xz,  then  the  coordinates  of  the  three  comers  of  the  base  will 
be 

0,  0,  2',  I  x",  0,  z",  I  x"\  y"\  2'\ 
and  the  equations  of  the  three  edges  terminating  in  the  vertex  will  be 


y  =  0 


X 

y  =  0 


X 

_2/" 


s» 


y  =  "^''- 


Now  the  perpendicular  distance  between  each  of  these  edges  and  the 
opposite  edge  of  the  base  will  evidently  be  equal  to  the  perpendicular 
-demitted  from  the  origin  on  a  plane  drawn  through  the  latter  edge* 
and  parallel  to  the  former.  Hence,  denoting  the  three  planes  through 
the  edges  of  the  base  by 

Arr  +  Bt/  +  Cz  =  1  I  Ex  +  Fy  +  Gz  =  1  I  la;  +  Ky  +  L2  =  1, 
they  must  be  drawn  so  as  to  fulfil  the  conditions  (See  Jlnal.  Geom.) 


Gz   =1 
Ea;"'+F«/"'+G2"'  =  l 

Ex"  +Gs"=0 


dz   =1 

Ax"  +C2"=1 

Ax""+By"'+Cz"'=0 

These  conditions  fix  the  following  values  for  A,  B,  C,  &c.,  viz 


Ix"  +Lz"=l 

Ix"'+Kj/"'  +  L2"'=l 

Lz'  =0 


-  A--4 

z  '  x" 


X  z 


X  y  z        X  y 

x"'z" 


y  z 


G  =  ir,  E   =   _-4-,,F=4  + 

z  x  z  y  X  y  z 

L  =  0, 1  =  4,  K 


y  z 


Hence,  calling  the  breadths  B,  B',  B",  we  have  {Anal.  Geom.) 


■  =  A='+B2+C2 


:E3+F2+G==: 


_  (y'V — y"'z"Y-\-{x"'z" — x"'z — x'V")^+  {x"y"y 
{x"y"'z'y 
{y"'z"Y-\-  {x'z  +  x"'z" — x"z"'y  +  {x"y"y 
{x'y'zf 


THE  DIFFERENTIAL  CALCULUS.  839 


4-,=  F+K=+L.-(!'-^'>'+(^"^-^'"^'>' 


B"3  {x"y"'z'y 

Hence  ^  +  ^  +  ^  = 

I  {z'—z"')x"+x"z"\^+  {y"'zy-{-  {x"—x"'yz'^  /   •  ^^  i/  ^  ^ ^'^ 

Again,  the  expressions  for  the  heights  or  perpendiculars  demitted 
from  each  of  the  points 

(0,0,0);  (0,0,/);  (x",  0,  s") ;  {x"\y"',  z'"), 
upon  the  plane  which  passes  through  the  other  three  are,  severally, 
{Anal.  Geom.) 

(z"  — z')-  y'"^  +  \  {z"  —  z)  x"  +  (2  —  z")  a;"'P  +  {y"'x"Y 


.. {^Y'^Y 


{z"y"'Y  +  {x"z"'  —  x"'z"Y  +  {x"y"'Y 

{y"'z'Y  +  {x"'z'f  {x'z'Y 

'   -L  +  JL  +  JL+     ' 

•   •     112     ^     Wa   ^    IT//2  T^ 


{z"—z'Yy""'+\{z"'—zf)x"+{z'—z")x"'\^+  \ 

2{3^yy+{y'''zy-^{x''2'''—x'''zfy-i-{y'''''-{-x'''''-\-x'''')z'''i 

■r- {x"y"'zfy . .  .  {2), 
which  expression  is  the  same  as  that  before  deduced,  and  thus  the 
theorem  is  established  by  a  process  purely  analytical.  This  remarka- 
ble property  was  discovered  by  JVT.  Brianchonj  and  formed  the  sub- 
ject of  the  prize  question  in  the  Ladies'  Diary  for  1830 :  a  solution 
upon  difierent  principles  may  be  seen  in  the  Diary  for  1831. 


END  OF  THE  DIFFERENTIAL  CALCULUS. 


NOTES 


Note  {A),  page  19. 

The  expressions  for  the  differentials  of  circular  functions  are  all 
readily  derivable,  as  in  the  text,  from  the  differential  expressions  for 
the  sine  and  cosine.  We  here  propose  to  show  how  these  latter  may 
be  obtained,  independently  of  the  considerations  in  art.  (14). 

By  multiplying  together  the  expressions 

COS.  A  +  sin.  A  %/ —  1,  cos.  Aj  +  sin.  Aj  V —  1, 
the  product  becomes 

cos-  A  cos.  Ai  —  sin.  A  sin.  Ai, 
=  (cos.  A  sin.  Ai  +  sin-  A  cos.  Ai)  V —  1- 
But  {Lacroix's  Trigonometry.^ 

cos.  A  cos  A,  —  sin.  A  sin.  Aj  =  cos.  (A  +  A,)  )  ,, . 

cos.  A  sin.  A,  +  sin.  A  cos.  Ai  =  sin.  (A  +  Aj)  |  *  '  *  w' 
hence  the  product  is 

cos.  (A  +  Ai)  +  sin.  (A  +  Ai)  %/—  1, 


=  cos.  A'  +  sin.  A'  ■s/—  1.* 
Consequently,  the  product  of  this  last  expression,  and 


cos.  Ajj  +  sin.  A2  -s/ —  1, 
is 

cos.  (A'  +  A2)  +  sin.  (A'  +  A2)  n/'^^oT,! 
=  cos.  A"  +  sin.  A"  y/—  1, 
the  product  of  this  last,  and 

COS.  A3  +  sin.  A3  \/ —  1, 
is 


COS.  A"'  +  sin.  A'"  v/—  1. 

*  Writing  A'  for  A  +  A ,.  +  Writing  A"  for  A'  -f  As  &c.  Ed. 

31 


242  NOTES. 


Hence,  generally, 
(cos.  AH- sin.  An/ —  l)(cos.Ai+sin.AiN/—  l)(cos.A2+sin.A2V--l) 


&c. 


cos.(A-|-Ai+A2+A3+&c.)sin.(A  +  A,  +  A2+A3+&c.)N/— 1. 

Supposing,  now, 

A  =  Ai  =  A2  =  A3  =  &c. 

this  equation  becomes 

(cos.  A  +  sin.  A  \/ —  1)"  =  cos.  nA  ±  sin.  nA  V —  1, 
or,  since  the  radical  may  be  taken  either  +  or  — 

(cos.  A  ±  sin.  A  V —  1)"  =  cos.  nA  ±  sin.  nA  \/ —  1, 
which  is  the  formula  o^  Demoivre,  n  being  any  whole  number. 

n 
Put  a  =  —  A  then 
m 


(cos.  a  ±  sin.  a  s/ — 1)"  =  cos.  ma  ib  sin.  ma  \/ —  1, 

=  COS.  nA  ±  sin.  nA  V  —  1  =  (cos.  A  ±  sin.  A  \f —  1)", 

therefore,  extracting  the  mth  root, 

n  n         

COS.  — A  ±  sin.  —  Av/  —  1  =  (cos.  A  ±  sin.  A  s/ — 1)"* 

which  is  the  formula  when  the  exponent  is  fractional. 

Having  thus  got  Demoivre^s  formula,  we  may  immediately  deduce 
from  it,  as  in  art.  (22),  the  series 

cos.  wA  =  cos.  "A — — -  cos."~^  A  sin.^  A  +  &c. 

sin.  nA=»  cos."-'  A  sm.  A ^ -^ ^cos."-'Asm.3A+  &c. 

Let  n  =  i,  sin.  A  =  0  =  A  .*.  nA  =  f  =  any  finite  quality  a:, 
hence,  by  these  substitutions,  the  foregoing  series  become 

cos.x=l-3-:^  +  p-^-3-^-&c. 

consequently, 

d  sin.  X  =  (1  —  - — -  +  :; —  —  &c.)  dx  =  cos.  xdar, 

\   t   a  1.^.0.4 


NOTES. 


243 


I  had  intended  to  have  given  here  another  method  of  arriving  at  the 
differentials  of  the  sine  and  cosine,  and  to  which  allusion  is  made  at 
page  41,  but,  upon  close  examination,  I  find  that  the  process  I  had 
then  in  view -is  liable  to  objection,  and  is  therefore  best  omitted. 


Note  {B),page  91. 
Demonstration  of  the  Theorems  of  Laplace  and  Lagrange. 

Let  it  be  required  to  develop  the  function 

u  =  Ys  where  z  =  F  (t/  +  xfz). 
By  differentiating  the  second  of  these  equations,  first  relatively  to 
•  tCj  and  then  relatively  to  y,  wc  have 

|  =  Fto+./.)(l+./'.|). 

Multiplying  the  first  by  -j-  and  the  second  by  ^,  and  subtracting, 

there  results 

dz       f  dz  _  dz  _      dz 

but  since  u  or  "Yz  depends  only  on  z,  we  shall  have 

du  _     1   dz  du ^   dz 

dx  dx^  dy  dy 

therefore,  eliminating  Y'z,  we  get 

du  dz      du  dz  _ 

dx  dy       dy  dx         ' 

dz 
or  putting  for  -7-  its  value  (1),  and  making  for  abridgment /z  =  Z, 

*  At  page  89  we  put/'z  to  represent  the  differential  coefficient  of/z  relatively 
to  *;  here  the  same  symbol  denotes  the  coefficient  relatively  to  z. 


72 


^-..  '^ri"vy^/,r 


244 


NOTES. 


dz 
this  last  expression  becomes  divisible  by  -r-  and  reduces  to 

dy 

^_ydu 

Tx-^d^j (^)' 

so  that  we  may  always  substitute  for  ~  the  quantity  Z  -j-. 

If  we  differentiate  the  preceding  equation  relatively  to  x,  we  shalli 
obtain 

du 
dZ-r 

<Pm  ^ ^  ....  (2), 

di?  dx       '  '  '  ' 

but  the  expression  Z  —  being  no  other  thanfz.Yz  — ,  that  is  to  say^ 

dz 
a  function  of  z  multiplied  by  — ,  we  may  consider  it  as  the  differential 

coefficict  of  some  new  function  of  2,  which  we  may  represent  by  «i» 
and  we  shall  then  have 

du 

dui  du  dy  <Pui 

=  Z  -7-,  and  ~ 


dy  dy^  dx  dxdy 

therefore  (2),  inverting  the  order  of  the  differentiations  in  this  last  ex- 
pression, 

dui 
d'u  dPui  dx 

dx^  dydx  dy       *'''<<)' 

now  it  must  be  observed  that  the  relation  (A)  exists,  whatever  be  the 
function  u ;  it  therefore  exists  for  the  function  Ui,  hence 
dui  _      dw, 
dx  dy ' 

Substituting  then  in  (3)  for  -p-  its  value  here  exhibited,  and  after- 
wards for  —r^  its  equal  Z  -7-,  there  results 
dy         ^  dy 


dui 
<P«  dy 

dx^    ~        dy 


(B). 


NOTES.  245 

Differentiating  this  last  equation  relatively  to  x,  we  shall  obtain 

dPu  dy 

dx^  dxdy  ' 

UAt 

and  considering,  as  before,  the  function  Z^  -r-  to  be  the  diflerential 

dy 

coefficient  of  some  new  function  of  2,  viz.  Wo,  we  shall  have,  by  invert- 
ing the  order  of  the  differentiations, 

^du<i 
dv?  _  y^du        fPtt     _        dx 
dy  dy         dx^  dy^      •  '  •  -  K  )^ 

but  the  equation  (A)  subsisting  for  every  function  m,  must  have  place 
for  the  function  u^-,  hence 


du^ 
dx 

therefore 

(4), 

du^ 
dx 

_  „3  du  d^u 

dy^  dsp 

cPZ^ 
dy' 

du 

dy 

.  .  (C). 

The  analogy  among  the  expressions  (A),  (B),  (C),  is  obvious,  and 
we  shall  now  show  that  this  analogy  continues  uninterrupted ;  that  is, 
generally,  if 

^22"-'  — 
d"~*M  _  dy 

d^  ~        dy^^~  •  •  •  *  ^^^' 
then 

dar  dy"-'       •  •  •  •  <.^^;- 

For  considering,  as  before,  the  function  Z"~'  -7-  to  be  the  differentia 

ay 

coefficient  of  some  new  function  of  2,  viz.  m„_„  so  that 


dy         dy 
we  have,  by  differentiating  (M), 


(5). 


246 


NOTES. 


dru  '    dx 


dx"  dy"~^      ' 

but  the  equation  (A)  subsisting  for  every  function  of  z  subsists  for 
«;_i  therefore 

du„_i  _      du^i  _ 
dx  dy    ' 

hence  (5) 

,„       dr-^Z"^      d-^{fz)^ 

d"u  _  dy  _  dy 

dxT  dy"-^  df-^       • 

Let  now  x  =  0  in  the  original  function,  and  in  each  of  the  coefficients 
du  d^u  d"u     ,  , 

Tx^d^"-'  d^'thenwehave 

[«]  =  W'.Fy  =  cpy,  [z-]  =  Fy  .-./C^]  =J\Fy  =  4^^ 
and 

Consequently,  by  Maclaurin's  theorem, 

d.i^yr^    _^ 

,    .     d^y    X   .         ^^^'    dy        oe' 


N3  d'py 


^^^^      dy  x"" 

d^'       rr^a  +  ^^- 

which  is  the  theorem  of  Laplace. 

The  preceding  investigation  is  taken  with  some  slight  variation 
from  the  large  work  of  Lacroix,  vol.  i.  p.  279. 
In  the  particular  case  where 

u  =  'Vz  and  z  =  y  -{■  xfz, 
we  have 

[«]  =  Wy,  [z]  =Fy  =  T,,/[z]  =-^j=fy: 
hence  the  development  is  then 


NOTES. 


S!47 


«='•'!/  +fy-3^-i  +  — ^—  ■  -.-2  +  «"=• 

which  corresponds  with  the  theorem  o£  Lagrange,  given  at  page  89. 


Note  (C),pag-e  127. 

Suppose  the  function /(a?  +  h)  fails  to  be  developable  according 
to  Taylor's  series  for  the  value  x  =  a,  and  let  the  true  development 
be  represented  by 

f{a+h)  =fa  +  A/ia  4-  /3/i«+^  +  C/i«+/?+y  +  &c (1), 

the  terms  being  arranged  according  to  the  powers  of  h.  It  is  requir- 
ed actually  to  find  these  terms. 

Let  the  difference/ (a  +  h)  — fa  be  divided  by  such  a  power  of 
h,  that  the  quotient  will  become  neither  0  nor  co  ;  when  h=  0,  such 

a  power  of  A  can  be  no  other  than  h  ,  or  that  which  ought  to  appear 
in  the  first  term  of  this  difference,  for  if  the  developed  difference  were 
divided  by  a  lower  power  of  h  than  this,  the  quotient  would  evidently 
be  0,  when  A  =  0,  emd  if  it  were  divided  by  a  higher  power,  the  quo- 
tient would  be  infinite  when  h  =  0;  hence  the  proper  divisor  h  be- 
ing found,  if  we  put  ^  =  0  in  the  quotient,  the  result  will  be  simply 
A ;  having  thus  found  the  true  first  term  of  the  difference,  let  it  be 
transposed  to  the  other  side,  and  we  shall  then  have  the  difference 

Now  the  first  side  of  this  equation  being  known,  we  have,  as  before, 
to  find  that  power  of  h,  that  it  may  be  divided  by,  so  that  the  quotient 

may  be  neither  0  nor  infinite,  when  h  =  0  this  power  will  be  w  ,  and 
putting  h  =  0  in  the  quotient,  the  result  is  B,  and  in  this  manner  it  is 
plain  that  all  the  terms  of  the  series  are  to  be  determined. 

Let  now  y  =fxhe  the  equation  of  a  plane  curve,  and  Y  =  Fa;  the 
equation  of  another  having  a  common  point  with  the  former,  at  which 
j;  =  a,  then 

Fo  =  /a ; 


248  NOTES. 

beyond  this  point  the  ordinates  of  the  two  curves  for  any  abscissa 
{a  +  h)  will  be/(a  +  h)  and  F  (a  +  h),  and  their  difference  D 
will  be 

D  =/(a  +  fe)  — F  (a  +  h). 
Let  us  now  develop  the  function  F  (a  +  ^),  as  we  have  done  the 
function/ (o  +  h),  and  we  shall  have 

F  (a  +  h)  =  Fa  +  X'h"-'  +  B'/i'^'+^'  +  C'/i'''+^''^' +  &c.  (2) 
and  it  may  be  shown  precisely,  as  at  page  128,  that  the  greater  the 
number  of  leading  terms  in  the  two  developments  (1),  (2),  are  the 
same,  the  nearer  will  the  developments  themselves  approach  to  iden- 
tity, so  that  no  curve  passing  through  the  point  common  to  the  two 
former  can  approach  so  closely  to  either  in  the  vicinity  of  that  point, 
unless  in  the  development  of  the  ordinate  the  same  number  of  leading 
terms  agree  with  those  in  (1). 

Lagrange  observes  {Theorie  des  Fonclions  Analytiques,  p.  184,) 
that  we  may  call  contact  of  the  first  order,  contact  of  the  second  or- 
der, &c.  the  approximation  of  two  curves,  for  which  the  two  first 
terms,  the  three  first  terms,  &c.  are  the  same  in  the  developments  of 
the  functions  which  represent  the  ordinates.  All  other  authors  who 
advert  to  this  subject  make  the  same  remark,  but  it  is  erroneous,  as  a 
simple  instance  will  show. 

Let  the  developed  ordinates  be 

A  +  B/i^  +  Clfi  +  Bli'     -t-  &c. 

K-\-Bh?  +  Ch^  +  M^iy   +  &c. 
According  to  Lagrange  the  contact  here  would  be  of  the  second  or- 
der, but  by  Taylor's  theorem,  these  developments  would  be 

A  +  Oh+BK"  ■}-  Oh''  +  Oh*  +  &c. 

A  +  O/i  +  B/i^  +  Oh""  +  Oh*  +  &c. 
and  therefore,  by  the  general  principles  established  in  Chap.  IL  Sect. 
II.  the  contact  is  of  the  fourth  order,  at  least.  If  in  the  true  develop- 
ments the  term  next  to  Oh*  were  the  same  in  each,  and  the  exponent 
of  /i  a  fraction  between  4  and  5,  while  the  term  following  differed  in 
the  two  series,  the  contact  might  be  properly  said  to  be  of  the  fifth 
order ;  but  the  sign  of  the  difference  of  the  two  developments,  when 
h  is  negative,  will  obviously  depend  on  the  fractional  exponents  of  /», 
in  the  terms  immediately  beyond  those  which  agree  in  the  two  series. 


NOTES.  249 

Note  (D),  pag-e  224. 

As  the  process  from  which  the  expressions  for  a,  ^,7,  given  in  ar- 
ticle 203,  are  deduced,  is  not  immediately  obvious,  we  shall  here  ex- 
hibit it  at  length  for  the  first  of  these  expressions. 

From  the  formulas  for  p"  and  q",  at  article  201,  we  immediately 
get  for  p'p"  +  q'q"  the  value 

{ePy)  jdy)  jda;)  —  jd'x)  {dyf  +  {(Pz)  (dz)  jdx)  —  {(Px)  {dzf 

{dxY 
^  (dx)  \  (c%)  jdy)  +  {drz)  {dz)  \  -  jd'x)  \  {dyf  +  {dzf  \ 

{dxf 
^  {dx)  \  {ds)  {d^s)  —  ((fx)  {dxf  I*  —  {dJ'x)  \  {dsf  —  {dxY  I 

{dxy 
_  (dx)  {ds)  (d's)  —  ((Fx)  {dsY  ^  {ds)  \  {dx)  {d^s)  —  {dl'x)  {ds)] 

{dxY  {dxy 

A 

Therefore,  putting,  for  brevity,  instead  of  the  denominator,  in 

the  expression  for  a  in  article  201,  we  have 

,    {dsf      {d'^x)  {ds)  —  {dFs)  (dr) 
-  =  ^+-J-'  ^. 

=  X  +  r^  -7—,  article  66, 
ds 

and  by  a  similar  process  the  expressions  for  {3  and  y  are  obtained. 

The  expressions  a  —  x,  (3  —  y,y  —  2  are  obviously  the  projec- 
tions of  the  radius  of  curvature  r  on  the  axes  of  x,  y,  z.  But,  if  we 
represent  the  inclinations  of  the  radius  to  the  axes  by  X,  fjt,  v,  the  ex- 
pressions for  the  projections  will  be 

r  cos.  X,  r  cos.  (x,  r  cos.  v, 

so  that  (206)  we  have,  for  the  angles  of  inclination,  the  values 

#x  d?y  (Pz 

COS.  X  =  r-r-^,  cos.  (u,  =  r-j^,  cos.  v  =  r^-^-. 
ds^  ds^  dsr 

By  employing  these  expressions  JVf.  Cauchy  has  arrived  by  rather  a 
novel  process  at  the  theorem  of  JMTeMsmer,  given  at  p.  181. 

*  Because  from 

(d5)«  =  {dxf  +  {dyf  +  {dzf 
we  get 

{da)  (d»»)  =  {dx)  {d?x)  +  {dy)  (rf»j,)  +  (dz{  {dh), 

32 


250 


NOTES. 


Let  the  equation  of  any  curve  surface  be 
u=F  {x,y,z)  =  0, 

upon  which  is  traced  any  curve  MG',  deter- 
mined by  the  equation 

y  =  9^» 

joined  to  the  preceding. 

If  through  the  tangent  MT  to  this  curve,  and  also  through  the  nor- 
mal MN  of  the  surface,  we  draw  a  plane,  we  shall  be  furnished  with 
a  normal  section  MG,  of  which  the  radius  of  curvature  r,  at  M,  will 
be  some  portion  of  the  normal  MN.  Also  the  radius  of  curvature  r' 
of  the  assumed  curve  MG',  at  the  same  point,  will  be  some  portion 
of  the  line  MN',  perpendicular  to  the  tangent  MT. 

Now,  considering  s  to  be  the  independent  variable,  we  have,  for 
the  incUnations  of  r  to  the  axes  the  expressions  above,  viz. 
,  d^x     ,  cFy     ,  <Fz 

and  the  inchnations  of  r  or  of  MN  to  the  axes  are  (127) 
du       du       du 
dx^      dij^      dz' 

Hence,  calling  the  angle  N'MN,  between  the  two  radii,  ij,  we  have 
{Anal.  Geom.) 

_      ,  ,  du    d^x  j^  du    d^y         du    d?z 
dx     ds^         dy     ds^         dz     ds' 
But  the  equation  of  the  surface,  considered  as  one  of  the  equations 
of  the  curve  MG',  gives  after  two  successive  differentiations,  still  re- 
garding s  as  the  independent  variable 

du     <Px    f^  du      ^y    ,    du      d?z  

dx       ds^         dy       ds^    '     dz       ds' 
d?u      dj?        d^u      d'xf        d?u      dz^ 
J"~    d^    ~d^~'df     ~d^~~d^    1^ 
d?u      dxdy  d?u       dxdz  d^u 


—  2 


dydz 


dxdy       ds^  dxdz       ds^  dydz      ds^  ' 

Now  whatever  be  the  curve  MG',  provided  only  its  tangent  MT  = 
x'  remains  unchanged,  the  second  member  of  this  last  equation  will 

remain  unchanged,  because  the  values  of  -j-,  ~-^  -r->  which  are  the 

di   as  ds 


NOTES.  251 

same  as  those  of  -j-j,  -y^,  -7-^,  remain  unchanged.       Therefore 

CtX        (JLX      0tX 

this  second  member  being  substituted  in  the  expression  for  cos.  w 
leads  to  a  result  of  the  form 

r  =  K  cos.  w, 

K  being  a  constant  expression  for  all  the  curves  on  the  proposed  sur- 
face which  touch  MT  at  the  point  M.  Put  now  in  this  expression 
cd  =0,  then  r'  becomes  r,  therefore 

r  =  K,  consequently  r  =  r  cos.  w, 
which  result  comprehends  the  theorem  of  JVlewsmer,  since,  if  the  curve 
MG'  is  plane,  its  plane  will  coincide  with  N'MT,  and  the  angle  w  of 
the  two  radii  will  become  the  angle  formed  by  the  plane  N'MT  of  the 
oblique  section  with  the  plane  NMT  of  the  normal  section  passing 
through  the  same  tangent  MT. — Leroy  Analyse  Appliquee  d,  la  Ge- 
ometne,  p.  268. 

We  may  take  this  opportunity  of  remarking  that,  in  our  investiga- 
tion of  this  theorem,  at  p.  182,  it  might  easily  have  been  shown,  with- 
out referring  to  article  86,  that 

-—  =  1  +  tan.2  6, 
dxr 

because,  by  the  right-angled  triangle, 

x"  —  x^  sec.^  d  =  a;^  (1  -f  tan.^  H) 

.'.  -7-:r  =  1  +  tan.^4  =  -—-. 
dar  dxr 


Note  (E),  page,  106. 

The  erroneous  doctrine  adverted  to  at  page  106  is  laid  down  also 
by  Lacroix,  in  his  quarto  treatise  on  the  Calculus,  vol.  1,  p.  340,  from 
whom,  indeed,  Mr.  Jephson  seems  to  have  adopted  it.  The  princi- 
ple as  stated  by  Lacroix  is  "  que  la  serie  de  Taylor  devient  illusoire 
pour  toute  valeur  qui  rend  imaginaire  Pun  quelconque  de  ces  terms  ; 
et  que  cela  peut  arriver  sans  que  la  fonction  soit  elle-meme  imagi- 
naire," It  is  very  remarkable  that  analysts  should  have  hitherto  held 
such  imperfect  notions  respecting  the  failing  cases  of  Taylor's  theo- 
rem. 


NOTES  BY  THE  EDITOR, 


Note  (A')  page  15. 

As  the  Algebra  here  referred  to  may  not  be  in  the  hands  of  the 
student,  we  shall  find  the  differential  coefficient  of  a  logarithmic  func- 
tion, by  previously  obtaining  that  of  an  exponential  one,  which  is  the 
course  pursued  by  most  writers  on  the  calculus.     Let 

u  =  a'  .  .  .  .  (1), 
in  which  if  a;  be  increased  by  h,  we  shall  have 
u'  =  a^  =  a"  X  aK 
Now  in  order  to  develop  the  last  factor  of  this  product,  we  suppose 
a  =  1  +  6,  in  order  to  subject  it  to  the  influence  of  the  Binomial 
Theorem,  we  shall  then  have 

ji       /,    I    ina       t    I   Li   I  '*'    ^ — 1     LI   I   h    h — 1     h  —  2 
t^={l  +  bf  =  l  +  hb  +  -.  —^  .b^^--.  — 2—  •  — 3— 

h'  +  &c. 
The  multiplication  indicated  in  the  second  number  of  this  being 
executed,  and  the  result  ordered  according  to  the  powers  of  h,  repre- 
senting by  sh^  the  sum  of  all  the  terms  containing  powers  of  h  above 
the  first,  we  shall  then  have 

6^        6^        6^ 
a*=  1  +  h{b  —  —  +  ___  +  &c.)  +  s/i2 

^  O  4 

Both  members  of  which  being  multiplied  by  a^,  designating  the  coeffi- 
cient of  h  within  the  parentheses  by  c,  we  shall  then  have 

a'  X  a''  =u'  =  {1  +  ch  +  sh?)  a'. 
The  primitive  function  being  taken  from  this,  leaves 

u'  —  M  =  ca'h  +  a'sh?, 
whence 

U  tt  ,  , 

— r —  =  ca'  -f-  a'sh. 


NOTES.  253 


which,  when  h  =  0  becomes 


^  =  00-....  (2), 


where 


1  2  3  4  ^  ^ 

It  is  thus  perceived  that  the  differential  coefficient  of  an  exponen- 
tial function,  is  equal  to  that  function  multiplied  by  a  constant  num- 
ber c,  which  is  the  above  function  of  its  base.  We  have  from  equa- 
tion (2), 

du  =  ca'dx, 
and  we  perceive  from  equation  (1)  that  log.  u  =  a;,  whence  d  log. 
u  =  dx'^  eliminating  dx  between  this  and  the  last,  we  have 

du  =■  ca'd  log.  M, 
and 

,,  du         I      du 

d  log.  u  =  —^  =  —  .—. 
ca"         c      u 

The  differential  coefficient  therefore  of  a  logarithmic  function  is 
equal  to  the  differential  of  the  function  divided  by  the  function  itself, 

multipUed  by  the  constant  — ,  the  modulus  of  the  system  whose  base 

is  a.     The  modulus  of  the  Naperian  or  Hyperbohc  system  of  Loga- 
rithms being  unity,  we  have 

d  .  lu  =  — , 
u 

lu  representing  the  Naperian  Log.  of «. 


NoTis  (B')  page  21. 

The  leading  part  of  article  15,  in  regard  to  the  notation  relative  to 
inverse  functions,  though  very  plausible,  is  nevertheless  calculated  to 
mislead  the  student.  For  in  the  equation  x  =  F~'  y,  expressing  the 
function  that  x  is  of  y,  the  direct  function  being  y  =  Fa:,  the  symbols 
F  and  F~'  should  not  be  considered  as  quantities  or  operated  upon  as 
such,  since  they  here  stand  in  place  of  the  words  a  function  of,  the 
forms  of  both  functions  being  different. 


254  NOTES. 

Note  (C)  page  65. 

Article  49  should  have  commenced  with  the  equation 
y  =  Far* 

and  though  the  succeeding  articles  are  full  and  ample  on  the  subject, 
it  may  not  be  amiss  to  present  the  maxima  and  minima  characteris- 
tics of  functions  in  less  technical  language. 

Remembering  the  note  page  63, 

let  u  =fx 
be  the  proposed  function  to  ascertain  whether  it  admits  of  maxima 
or  minima  values  ;  and  if  so,  by  what  means  they  and  the  variable  on 
which  they  depend  may  be  discovered. 

In  the  proposed  function  if  the  variable  x  first  increase  and  then 
decrease  by  any  quEuitity  h,  we  shall  then  have 

u  =fx  ....   (1), 
and  by  Taylor's  Theorem, 

,       ^,     ,  ,,  ,  dull   ,   dFu     h""      ,   dhi       h^       ,    d'u 


1.2.3.  4 


+   &c (2), 


"  —  ff         j\  —  du  h       d^u     Ir  d?u        ¥        .    d'^u 

u   -J{x  —  li)-u  —  —--Jr-^Y72~d^  17273  "^"d^ 

^'  +   &c.' (3). 


1.2.3.4 

Now,  in  order  that  the  function  u  under  consideration  may  attain  a 
maximum  or  minimum  value  (2)  and  (3),  must  be  both  less,  or  both 
greater  than  (1),  and  as  h  may  be  assumed  so  small  that  the  term 
containing  its  first  power  may  be  greater  than  the  sum  of  all  the  suc- 
ceeding terms,  (2)  will  be  greater  than  «,  while  (3)  will  be  less. 
Since  the  first  differential  coefiicient  has  different  signs  in  the  two 
developments,  the  function  therefore  cannot  attain  maxima  or  mini- 
ma values,  unless  this  coefficient  becomes  zero.     The  roots  of  the 

equation  -p  =  0,  will  give  such  values  of  x  as  may  render  the  func- 
tion a  maximum  or  a  minimum;  such  values  of  the  variable  being 


NOTES.  255 

substituted  in  the  second  differential  coefficient  -j-j-  if  these  render  its 

value  any  thing,  we  are  certain  the  function  may  become  a  maxi- 
mum if  that  value  is  negative,  or  a  minimum  if  it  be  positive  ;  for  in 
the  first  case  (2)  and  (3)  are  both  less  than  ii,  and  in  the  second  they 
are  both  greater.  But  if  the  same  values  of  x  render  the  second  dif- 
ferential coefficient  zero,  as  well  as  the  first,  we  readily  see  that  the 
third  differential  coefficient,  must  also  become  zero,  in  order  that  the 
function  may  admit  of  maxima  or  minima  values  :  because  this  coef- 
ficient has  different  signs  in  (2)  and  (3),  we  then  substitute  the  same 
values  of  a:  in  the  fourth  differential  coefficient,  which  has  the  same 
sign  in  (2)  and  (3),  if  these  render  it  negative  we  shall  have  a  maximum 
value  of  the  function,  and  if  positive  a  minimum  value ;  but  should 
this  coefficient  also  vanish  with  the  preceding  ones,  the  next  must  be 
examined,  and  so  on. 

In  order  therefore  to  determine  the  values  of  x,  which  render  the 
proposed  function  a  maximum  or  minimum  we  must  find  the  roots  of 

the  equation  —  =  0,  and  substituted  then  in  the  succeeding  differen- 
tial coefficients,  until  we  find  one  that  does  not  vanish ;  if  this  be  of  an 
odd  order,  the  roots  we  have  employed  will  not  render  the  function  a 
maximum  or  minimum,  but  if  it  be  of  an  even  order,  then  if  this  coeffi- 
cient be  negative  we  have  a  maximum  value  of  the  function,  but  if  posi- 
tive a  minimum  value. 


THE  END. 


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